A note on matrix volume preservers

Let $\mathbb{R}^{m \times n}$ be the vector space of $m \times n$ real matrices, and let $\phi\colon \mathbb{R}^{m \times n} \longrightarrow \mathbb{R}^{m \times n}$ be a linear transformation such that $\operatorname{vol}(\phi(A)) = \operatorname{vol}(A)$ for all $A \in \mathbb{R}^{m \times n}$. If $m \neq n$, then there exist two orthogonal matrices $P \in \mathbb{R}^{m \times m}$ and $Q \in \mathbb{R}^{n \times n}$ such that $\phi(A) = P A Q$ for all $A \in \mathbb{R}^{m \times n}$. If $m = n$, then there exist two orthogonal matrices $P \in \mathbb{R}^{n \times n}$ and $Q \in \mathbb{R}^{n \times n}$ such that either $\phi(A) = P A Q$ for all $A \in \mathbb{R}^{n \times n}$ or $\phi(A) = P A^{\mathrm T} Q$ for all $A \in \mathbb{R}^{n \times n}$.