Q: pair of orthonormal vectors map to any other pair of orthonormal vectors by element of $SO(n)$ Let $v_1,v_2$ be orthonormal vectors in $\mathbb{R}^n$. Now let $w_1,w_2$ be any other pair of orthonormal vectors in $\mathbb{R}^n$. (Assume all non zero). I want to show that there is $A\in SO(n)$ such that $gv_i=w_i$. 
Attempt: 
To show this it is enough to show that there is an element in $SO(n)$ that sends the standard basis vectors $e_1,e_2$ to $v_1,v_2$. I think the matrix that has $v_1$ as the first column and $v_2$ as the column will do this. But I do not know how to show that this is in $SO(n)$.
Is this the correct approach?

A: If $n=2$, this may not be true. (Consider $v_1 = (1,0)$ and $v_2 = (0, -1)$.)
However if $n \ge 3$, then you may consider a matrix with $v_1$ and $v_2$ as the first two columns, and then fill in the other columns by completing an orthonormal basis. This gives you an element of $O(n)$ that sends $e_1 \mapsto v_1$ and $e_2 \mapsto v_2$. If this element is not in $SO(n)$, then, say, replacing the last column with its negation will fix it.
