Let $S^n$ be the n-dimensional unit sphere in $\mathbb{R}^{n+1}$.
Let $\omega$ be the volume form that's defined for all $p\in M$ by $\omega_{\vec{p}} = x^1\wedge...\wedge x^n$ s.t. $x^1,...,x^n$ is the dual basis for some orthonormal directed basis $x_1,...,x_n$ of $T_{\vec{p}}(M)$ .
I would like to show that $\int_{S^n}\omega \neq 0$.
To the best of my understanding, not only does $\int_{S^n}\omega \neq 0$, but $\int_{S^n}\omega = Vol(S^n)$; However, I don't know how to prove neither claim.