Existence of Hodge Dual: obvious or non-trivial?

[nonequilibrium]

Some sources I have checked define the Hodge dual of a form $\omega \in \Omega^p$ as the object such that $\forall \eta \in \Omega^p: \eta \wedge \omega^\star = g(\eta,\omega) \textrm{ Vol}$ (where "Vol" is a chosen volume form).

I can see that there can be only one form with such a solution (i.e. unicity), but I can't see existence: how do we know there is such a form $\omega^\star \in \Omega^{n-p}$ that satisfies that definition?

(I can kind of see it locally using some basis argument, but how make it global...) The main thing I'm confused about is whether it should be obvious that it exists (since my sources don't give extra arguments), or whether it requires further justification.

 

[jgens]

Basically just construct it point-wise. Since forms are sections of tensor bundles evaluating at a point reduces the existence problem to vector spaces. Only a smoothness argument remains and for this local frames come to the rescue.