= Solution
The required actions have already been identified:
$$
\boxed{Mu_1=\mu_1v_1,\qquad Mv_1=-\mu_1u_1.}
$$
In three dimensions choose a unit <vector> $w$ perpendicular to both. Its one-dimensional <orthogonal complement> to the pair is invariant under $M$. On a one-dimensional real space, a <skew-symmetric matrix> must act as zero: $Mw=\eta w$ would imply $\eta=w^TMw=0$. Thus $Mw=0$.
The columns of $W=(u_1\ v_1\ w)$ form an <orthonormal basis>, so $W$ is an <orthogonal matrix>. The displayed actions give
$$
W^TMW=\begin{pmatrix}0&-\mu_1&0\\\mu_1&0&0\\0&0&0\end{pmatrix}=B,
$$
and therefore
$$
\boxed{M=WBW^T.}
$$
If $M=0$, take $\mu_1=0$ and any <orthogonal matrix> $W$. Otherwise its nonzero two-dimensional block supplies the pair used above.
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