There is a following exercise in my text:
Let $S^n$ be $n-$ dim sphere in $R^{n+1}$ with inclusion function $i:S^{n}\to R^{n+1}$. Let $\omega=\sum_{i=1}^{n+1}(-1)^{i-1} x_i dx_1 \wedge... dx_{i-1}\wedge dx_{i+1}\wedge ... \wedge dx_{n+1}.$ Prove that $i^*\omega \in \Omega^n(S^n)$ is Riemannian volume form on $S^n$.
I treied to manually compute this expression and the one which uses definition of Riemannian volume form when they act on some vectors in $T_xS^n$ but things gets complicated when $n$ is large and involves sum of matrix determinants which I don't know how to resolve. I managed to prove the result for small values of $n$.
How would you go with the general case?