Unfortunately, John Lee leaves it as an exercise to prove that the Hodge operator can be defined by $\omega\wedge*\eta=\langle\omega,\eta\rangle\mathrm{d}V_g$, where $\mathrm{d}V_g$ is the Riemannian volume form.
I searched for a proof online (and in Tu's books), but it seems like most courses on Differential geometry (DG) don't cover the Hodge operator. A proof or a reference that covers the Hodge Operator would be much appreciated. Since I am beginner in DG, I'd like to have a very detailed reference, similar to John Lee's books.