I am doing some problems in Spivak's calculus on manifolds, and I was stuck on this one. It says that:
If $w_1, ... ,w_{n-1} \in \mathbb{R}^n$, show that $$| w_1 \times ... \times w_{n-1}| = \sqrt{\det(g_{ij})}$$ where $g_{ij} = \langle w_i, w_j \rangle$.
There is a hint that asks for us to apply Problem 4-3 to a "certain" $n-1$ dimensional subspace of $\mathbb{R}^n$, where problem 4-3 basically says that, given the volume element $\omega \in \Lambda^n(V)$ determined by $T$ and $\mu$ with $w_1, ... ,w_n \in V$, then $$| \omega(w_1, ... , w_{n})| = \sqrt{\det(g_{ij})}$$ where $g_{ij} = T(w_i, w_j)$.
The problem that I am having with this question is that it's not clear how the hint actually helps us. I was thinking, it seems we should be taking $T$ to be the usual inner product, but then in order to apply 4-3 we need to choose an orientation, and I'm not sure how we can do that. Moreover, I'm not sure where the volume element is coming in, because it doesn't seem to be encountered in the actual problem itself. I'm also wondering what this $n-1$ dimensional subspace is supposed to be.
I was able to write out the norm of the cross product itself as a determinant (by definition), but I can't see any further reductions from there. Any advice on how to proceed?