= Solution
The <real tautological line bundle> has total space $E=\{(\ell,v):\ell\in\mathbb{RP}^n,\ v\in\ell\}$ and projection $(\ell,v)\mapsto\ell$. On the open chart $U_i=\{[x]:x_i\ne0\}$, the vector $s_i([x])=x/x_i$ is a continuous nowhere-zero <section of a vector bundle>, independent of the representative of $[x]$. The explicit <local trivialization> is
$$
U_i\times\mathbb R\longrightarrow E|_{U_i},\qquad ([x],t)\longmapsto([x],t s_i([x])),
$$
with inverse $([x],v)\mapsto([x],v_i)$. Both maps are continuous and linear on each fiber. The charts cover the base, proving the existence of the required <local trivializations>.
Let $E\to B$ be a rank-$r$ <vector bundle> oriented over a <commutative ring> $R$, with $r\geq1$. An $R$-<orientation> is a coherent choice of generator of each fiber's top <relative cohomology>, or equivalently a <Thom class> $u_E$. The <Thom isomorphism theorem> identifies $H^{q-r}(B;R)$ with $H^q(D(E),S(E);R)$ by $a\mapsto\pi^*a\smile u_E$. The <disk bundle> retracts to $B$, and the relative-to-absolute map becomes multiplication by the <Euler class> $e(E)$. The pair's long <exact sequence> therefore becomes the <Gysin sequence of a sphere bundle>:
$$
\cdots\longrightarrow H^{q-r}(B;R)\xrightarrow{\smile e(E)}H^q(B;R)\xrightarrow{p^*}H^q(S(E);R)\longrightarrow H^{q-r+1}(B;R)\xrightarrow{\smile e(E)}H^{q+1}(B;R)\longrightarrow\cdots.
$$
With real coefficients, the bundle must be oriented in the usual sense. For the computation here use $R=\mathbb F_2$, over which every real <vector bundle> is oriented, including the nonorientable <real tautological line bundle> when $n\geq1$.
The unit <sphere bundle> of $\gamma_n$ is $S^n$: a unit vector determines its line. Put $a=e_2(\gamma_n)=w_1(\gamma_n)\in H^1(\mathbb{RP}^n;\mathbb F_2)$. For $n\geq1$, the pullback $H^0(\mathbb{RP}^n;\mathbb F_2)\to H^0(S^n;\mathbb F_2)$ is an <isomorphism>, so the next <connecting homomorphism> is zero and multiplication by $a$ is injective on $H^0$. Since the <sphere>'s <cohomology> vanishes strictly between degrees zero and $n$, exactness makes multiplication by $a$ an <isomorphism> $H^{q-1}\to H^q$ for $1\leq q<n$, and injective for $q=n$. For $q=1$ these assertions also use the just-noted vanishing of the <connecting homomorphism> from $H^0(S^n)$.
The standard $n$-dimensional <CW complex> structure of <Real projective space> gives $H^q=0$ for $q>n$. Consequently the segment $H^n(S^n;\mathbb F_2)\to H^n(\mathbb{RP}^n;\mathbb F_2)\xrightarrow{\smile a}H^{n+1}(\mathbb{RP}^n;\mathbb F_2)=0$ makes $H^n$ at most one-dimensional. The preceding injectivity makes it exactly one-dimensional, generated by $a^n$. This includes $n=1$; $n=0$ is a point separately. We have derived the multiplication, not just the dimensions:
$$
\boxed{H^*(\mathbb{RP}^n;\mathbb F_2)=\mathbb F_2[a]/(a^{n+1}),\qquad |a|=1.}
$$
An <odd map between spheres> descends to $\bar f:\mathbb{RP}^n\to\mathbb{RP}^m$. The <odd maps pull back the real tautological line bundle> argument gives $\gamma_n\cong\bar f^*\gamma_m$: with $x$ a unit vector the fiber map is $t x\mapsto t f(x)$, unchanged when $(x,t)$ is replaced by $(-x,-t)$. Naturality of the first <Stiefel–Whitney class> gives $\bar f^*b=a$. If $n>m$, the relation $b^{m+1}=0$ would imply $a^{m+1}=0$ in a ring where this power is nonzero. Hence $\boxed{n\leq m}$. The case $n=0$ is immediate, and the same argument rules out $m=0<n$.
For the map with separate oddness, pass to $\bar g:\mathbb{RP}^n\times\mathbb{RP}^n\to\mathbb{RP}^n$. Let $a,b$ be the degree-one classes from its two domain factors and $c$ the target class. There is an <isomorphism>
$$
\bar g^*\gamma_n\cong p_1^*\gamma_n\otimes p_2^*\gamma_n,
$$
given on fibers by $(s x)\otimes(t y)\mapsto st g(x,y)$; flipping either unit representative leaves this map well defined. The <First Stiefel–Whitney class of a tensor product of real line bundles> gives $\bar g^*c=a+b$. By the <Künneth theorem>,
$$
H^*(\mathbb{RP}^n\times\mathbb{RP}^n;\mathbb F_2)=\mathbb F_2[a,b]/(a^{n+1},b^{n+1}).
$$
Pulling back $c^{n+1}=0$ shows that $(a+b)^{n+1}=0$. The two end monomials already vanish, while every interior monomial $a^j b^{n+1-j}$, $1\leq j\leq n$, is a distinct nonzero <basis> element. Therefore all interior <binomial coefficients> in row $N=n+1$ are even. To identify such rows, write $N=\sum_{i\in I}2^i$. In $\mathbb F_2[t]$ repeated squaring gives $(1+t)^N=\prod_{i\in I}(1+t^{2^i})$. If $I$ has at least two elements, the coefficient of $t^{2^{\min I}}$ is one and its exponent is strictly between zero and $N$, a contradiction. Thus $N$ is a power of two, as in <binomial coefficients with even interior terms>, and
$$
\boxed{n=2^k-1\quad\text{for some }k\geq0.}
$$
This is the necessary <cohomological obstruction to separately odd sphere multiplication>; it does not assert existence in every dimension of this form.
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