Does SO(3) preserve the cross product? Let $g\in SO(3)$ and $p,q\in\mathbb{R}^3$. I wondered whether it is true that $$g(p\times q)=gp\times gq$$
I am not sure how to prove this. I guess I will use at some point that the last row $g_3$ of $g$ can be obtained by $g_3=g_1\times g_2$.
But I assume there is an easier proof than writing everything out.

You may use the scalar triple product formula $r \cdot (p\times q)=\det(r,p,q)$ to prove that
$$
gr \cdot (gp\times gq)=gr \cdot g(p\times q)\tag{1}
$$
($=\det(r,p,q)$) for any vector $r$. Since $g$ is invertible, if $(1)$ holds for every vector $r$, we must have $gp\times gq=g(p\times q)$.
