Let $n\geqslant 1$ and $\mathbb R^n$ be a $n$-dimensional space.
Let $(u_1,\ldots,u_d)$ be $d$ vectors of $\mathbb R^n$, with $d\leqslant n$.
Then $(u_1,\ldots,u_d)$ delimit a parallelogram $P$ of dimension $d$ in $\mathbb R^n$.
What is the volume of $P$, considering $P\subseteq\mathrm{Span}(u_1,\ldots,u_d)$?
What I tried.
If $d=n$, the problem is easy, since
$$\mathrm{volume}(P)=\det U$$ where $M:=(u_1\vert\cdots\vert u_d)$ the matrix whose columns are $u_1,\ldots,u_d$.
I have no idea how to tackle this problem where $d<n$. Maybe we can use sub-determinants of size $d\times d$ of $U$, but I can not figure it out.