Let $n \ge 3$ and let $\pi$ be the natural representation of $SO(n)$ in $\Bbb C^n$ ( where by $SO(n)$ I mean the real group). Let $v_1, v_2$ be a pair of orthonormal vectors in $\Bbb R^n$ and $w_1, w_2$ a second such pair.
Show that $\exists g \in SO(n)$ such that $gv_i = w_i$ for $i = 1, 2$
My attempt:
So if $V= \Bbb C^n$ consists of column vectors, and since the group consists of matrices, I guess the natural representation is the one consisting in left multiplication by an element of the group? $\pi(g)v = gv$. Now my idea is that given the orthonormality of each set of vectors, in each case they can be part of a basis of $\Bbb R^n$, but I am not sure if that is helpful and how to proceed from here.
$v_1, v_2$ can be completed to an orthonormal basis (of $\Bbb{R}^n$) $v_1, v_2, \ldots, v_n$. Now put the $v_i$ as column vectors in a matrix $V$. Then $V\in O(n)$. If $\det(V)=-1$ you can invert the sign of one of the vectors $v_3,\ldots v_n$ to get a matrix $V\in SO(n)$. (This is where you use that $n\ge 3$). The matrix $V$ multiplying column vectors from the left maps $e_1, e_2$ to $v_1, v_2$ respectively (where $e_i$ are the standard basis of $\Bbb{R}^n$.)
Now repeat this process to get a matrix $W\in SO(n)$ such that $W$ multiplying column vectors on the left maps $e_1, e_2$ to $w_1, w_2$ respectively.
Finally let $g=WV^{-1}=WV^\top$.