Volume of $(n-1)$- simplex in $n$-dimension.

Question

This post gives a general way to calculate $k$-simplex in $n$-dimensional space with $k\leq n$. My question is, if $k=n-1$ and give vertices $v_{0}, \cdots, v_{n-1}$ are linearly independent, can we show that the simplex $S$ generated by $v_{0}, \cdots , v_{n-1}$ has a volume $Vol(S) = \frac{1}{n!}\det\begin{pmatrix}{\bf v}_{0} & \cdots & {\bf v}_{n-1}\end{pmatrix}$?

Answer 1

I think this is a direct consequence of the Cayley-Menger determinant.