If $v_1, \ldots, v_p$ are chosen uniformly within a unit sphere in $\mathbb{R}^n$, how to find $E(Vol(v_1, \ldots, v_n)^{2a}))$ for positive $a$?

Question

Suppose that vectors $v_1, \ldots, v_p$ are chosen independently and uniformly within a unit sphere in $\mathbb{R}^n$. I would like to find:

$$ E\left(\left[Vol(v_1, \ldots, v_n)^{2a}\right]\right) $$ for positive integer $a$.

That is, I'd like to find the expectation of the volume of the vectors taken to the power of $2a$, where $a$ is a positive integer. Is there an easy way to do this with integral jacobians?