= Solution
Let $D^n_+$ and $D^n_-$ be the two closed hemispheres of the <sphere>. Use $\gamma$ as the <clutching function> between the trivial bundles on these hemispheres. The <clutching construction> defines
$$
V_\gamma=
\bigl((D^n_+\times\mathbb R^k)\sqcup(D^n_-\times\mathbb R^k)\bigr)/\sim,
\qquad (u,v)_+\sim(u,\gamma(u)v)_-,\quad u\in S^{n-1}.
$$
The projection to $S^n$ is locally a <vector bundle>. In a collar of the equator, extend the transition function by keeping it constant in the normal coordinate; this gives explicit <vector bundle trivializations> across the seam. Since the transition matrices lie in the <special orthogonal group>, the standard fiber <orientations> and Euclidean <inner products> agree on overlaps. Thus this is an oriented rank-$k$ <vector bundle> with a <fiber metric>.
For $k\ge2$, choose a base point $u_0$ on the equator. A constant change of the negative hemisphere frame replaces $\gamma$ by $\gamma(u_0)^{-1}\gamma$, so we may suppose $\gamma(u_0)=I$. With $e_k$ the last standard unit vector, define the <section obstruction for a clutched vector bundle>
$$
\boxed{f_\gamma=[u\longmapsto\gamma(u)e_k]\in\pi_{n-1}(S^{k-1});\qquad r=n-1,\ s=k-1.}
$$
The crucial point is that a <nowhere-zero section> can be normalized using the <fiber metric>. It is then represented by continuous maps $\sigma_\pm:D^n_\pm\to S^{k-1}$ satisfying
$$
\sigma_-(u)=\gamma(u)\sigma_+(u)
$$
on the equator.
If $f_\gamma=0$, the <extension-null-homotopy criterion for a sphere> extends $u\mapsto\gamma(u)e_k$ to $D^n_-$. Take this extension as $\sigma_-$ and the constant map $e_k$ as $\sigma_+$. They glue to a <nowhere-zero section>.
Conversely, suppose such a section exists. Restricting a contraction of $D^n_+$ shows that $\sigma_+|_{S^{n-1}}$ is homotopic to a constant. The target <sphere> is path connected, so the constant can be moved to $e_k$. Multiplying this homotopy pointwise by $\gamma(u)$ gives a <homotopy> from $\gamma\sigma_+$ to $\gamma e_k$. The first of these maps equals $\sigma_-|_{S^{n-1}}$ and extends over $D^n_-$, hence is <null-homotopic>. So $\gamma e_k$ is <null-homotopic> too. A freely null-homotopic based sphere map represents zero in its <homotopy group>; any change of base point sends the zero class to zero. We have proved both directions:
$$
\boxed{f_\gamma=0\quad\Longleftrightarrow\quad V_\gamma\text{ has a nowhere-zero section}.}
$$
For $n=1$, the same argument reads as the extension criterion for a two-point boundary and uses the trivial set $\pi_0(S^{k-1})$ when $k\ge2$. For $k=1$, the <special orthogonal group> $SO(1)$ is trivial, so the <vector bundle> is already a <trivial vector bundle>; the corresponding constant map to $S^0$ has zero obstruction. These cases require no higher <homotopy group> calculation.
If $k>n$, then $n-1<k-1$ and the permitted vanishing of lower <homotopy groups> of a <sphere> makes $f_\gamma=0$. Normalize the resulting <nowhere-zero section> to a unit section $\sigma$, and set
$$
V'_x=\{v\in(V_\gamma)_x:\langle v,\sigma(x)\rangle=0\}.
$$
Local orthonormal frames show that these <orthogonal complements> form a rank-$(k-1)$ <vector bundle>. The explicit <vector bundle isomorphism>
$$
V'\oplus(S^n\times\mathbb R)\longrightarrow V_\gamma,
\qquad (x,v,t)\longmapsto v+t\sigma(x)
$$
has inverse $w\mapsto(w-\langle w,\sigma\rangle\sigma,\langle w,\sigma\rangle)$. Orient $V'$ so that a positive frame followed by $\sigma$ is positive in $V_\gamma$. This is <splitting a trivial line from a nowhere-zero section> and proves the oriented splitting
$$
\boxed{V_\gamma\cong V'\oplus\mathbb R\qquad(k>n).}
$$
Finally consider the <special orthogonal sphere fibration>
$$
SO(s)\longrightarrow SO(s+1)\xrightarrow{p}S^s,
\qquad p(A)=Ae_{s+1}.
$$
The fiber over $e_{s+1}$ consists exactly of $\operatorname{diag}(B,1)$ with $B\in SO(s)$. Near any unit vector, the <Gram-Schmidt process> continuously completes it to an oriented orthonormal frame, giving local sections of $p$ and local product <fiber bundle> charts. In particular $p$ has the <homotopy lifting property>.
Let $\alpha:S^r\to SO(s+1)$ be based and suppose $r<s$. The composite $p\alpha$ is a based <null-homotopic map> by the given vanishing of $\pi_r(S^s)$. Lift a based null-homotopy of $p\alpha$, beginning at $\alpha$, to a homotopy $G:S^r\times[0,1]\to SO(s+1)$. The lift might initially move its base point $u_0$, but $G(u_0,t)$ always fixes $e_{s+1}$. Right multiplication by $G(u_0,t)^{-1}$ corrects this: the homotopy
$$
\widetilde G(u,t)=G(u,t)G(u_0,t)^{-1}
$$
still covers the same sphere homotopy, remains based, and starts at $\alpha$. At its endpoint every matrix fixes $e_{s+1}$, so $\widetilde G(-,1)$ lies in the included $SO(s)$. Thus every <homotopy class> in the target is represented by one from $SO(s)$:
$$
\boxed{\pi_r(SO(s))\longrightarrow\pi_r(SO(s+1))\text{ is surjective for }r<s.}
$$
This proves <surjectivity of special orthogonal stabilization below the sphere dimension>. For $r=0$, this is the corresponding assertion about path components; the positive-rank <special orthogonal groups> are connected. The lifting argument also explains the zero term $\pi_r(S^s)$ in the <long exact sequence of homotopy groups of a fibration>.
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