In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said that since $\omega$ is non-degenerate, the $n$-fold wedge-sum $\omega \wedge \cdots \wedge \omega$ is a volume form and if $M$ would be a closed manifold $M$, then $\int_M \omega \wedge \cdots \wedge \omega \neq 0.$
Could anybody explain to me where these two conclusion actually come from, i.e. I see that the wedge sum is a top-degree volume form, but I don't see why this means that it is nowhere vanishing and especially why the integral is non-zero.