If $S$ is the simplex defined by the convex hull of $0$ and vectors $a_1, \ldots, a_p$, how show $Volume(S) = \pm \frac{1}{n!} det(a_1, \ldots, a_p)$?

Question

If $S$ is the simplex defined by the convex hull of $0$ and vectors $a_1, \ldots, a_p$, I would like to show $Volume(S) = \pm \frac{1}{n!}det(a_1, \ldots, a_p)$. In other words, the volume of the simplex is plus or minus the determinant of a matrix $A$ which has $(a_1, \ldots, a_p)$ as its columns.

My technique is to recognize that $S = \{v:v=\sum_{i=1}^pc_ia_i, \ c_i \geq 0, \ \text{and} \sum_{i=1}^p \leq 1\}$.

But, from here I am not sure what to do. Does anyone have any ideas?