I was reading the book "Differential forms and applications" by M. Do Carmo. In his proof of the divergence theorem as a corollary of Stokes theorem. He claims(without proof) the following
Given $\omega$ a 1-form in $\mathbb{R}^3$, and $\{e_1,e_2,N\}$ is a positive oriented basis with $N$ normal to $e_1$ and $e_2$. Then $\star\omega(e_1,e_2)=\omega(N)$
Now, I can check this by straight-forward computation, but this quite unsatisfying, as it doesn't shed any light on why this is true.
So, this is my question: What is the relation between the image of a k-form $\omega$ in a space $W$ and the image of $\star\omega$ in $W^{\bot}$?
Furthermore, is it true that:
Given a $\{ e_1,\cdots,e_n \}$ an oriented basis(that is $\nu(e_1,\ldots ,e_n)=\mbox{det}(e_i)>0$) and a k-form $\omega$. Is it true that, $\omega(e_1,\ldots,e_k)=\star \omega(e_{k+1},\ldots,e_n)$ ?
I've tried some cases and it seems to be true, however my calculations seem difficult to generalise to the general case. Any idea on whether this is true, an if it is, how to prove it?