The determinant $\det(A)$ is an alternating multilinear form in the columns of $A$. Given vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n\in\mathbb{R}^n$, we can interpret $|\det(A)|$ as the volume of the parallelotope generated by $\mathbf{a}_1,\ldots,\mathbf{a}_n$. More precisely, $\det(A)$ is the ''signed volume'' of the parallelotope.
Now consider $k$ vectors $\mathbf{a}_1,\ldots,\mathbf{a}_k\in\mathbb{R}^n$ and let $A$ be the $n\times k$ matrix with these columns. Then $\sqrt{\det(A^TA)}$ is the $k$-volume of the $k$-parallelotope generated by $\mathbf{a}_1,\ldots,\mathbf{a}_k$. But I have no idea how to define the corresponding ''signed $k$-volume''.
Problem: The function $\sqrt{\det(A^TA)}$ (assuming positive real square root) is not multilinear in the columns of $A$. It can be made multilinear by choosing an appropriate sign $\pm\sqrt{\det(A^TA)}$, depending somehow on the matrix $A$, but I don't know how to choose the sign in a systematic way. In the square case, the correct sign is just the sign of $\det(A)$.
More generally, I guess I want to understand the volume form on a $k$-submanifold of $\mathbb{R}^n$ induced by the standard (determinant) volume form on $\mathbb{R}^n$.