The standard definition of the Hodge star is as follows: the Hodge dual of a differential $p$-form defined on an $n$ dimensional manifold $M$, $\alpha \in \Omega^p(M)$, is the unique form $\star \alpha \in \Omega^{(n-p)}(M)$ that satisfies \begin{equation} \beta \wedge \star \alpha = \langle \beta, \alpha \rangle \omega \end{equation} for any differential $p$-form $\beta$. Of course, implicit in this definition, there is a choice of metric on $M$, which induces an inner product on $p$-forms, as well as a choice of volume form $\omega$, although from what I gather this tends to be taken to be the volume form induced by the metric. The Hodge dual is then an isomorphism $\star \colon \Omega^p(M) \to \Omega^{n-p}(M)$, which reflects the fact that these spaces have the same dimension.
I noticed, however, that really the Hodge star is built out of two steps: the choice of a volume form gives you an isomorphism between $(n-p)$-forms and $p$-vector fields, and the choice of a metric gives an isomorphism between $p$-vector fields and $p$-forms.
I worked through the details in the case $p=1$. First, given a volume form $\omega$, we have what one could call a primitive Hodge star, $\ast \colon \Gamma(TM) \to \Omega^{n-1}(M)$ which is just an interior product: $\ast X = \iota_X \omega$. The other isomorphism is just the musical isomorphism $\sharp \colon \Omega^1(M) \to \Gamma(TM)$, which depends on the choice of a metric of course. I claim that the composition of the two is the Hodge star, namely, $\star \alpha = \ast \sharp \alpha = \iota_{\sharp \alpha} \omega$. And indeed, using the fact that the interior product is a derivation I can show this, since for any 1-form $\beta$ \begin{align} \beta \wedge (\ast \sharp \alpha) &= \beta \wedge (\iota_{\sharp \alpha} \omega) \\ &= (\iota_{\sharp \alpha} \beta) \wedge \omega - \iota_{\sharp \alpha}(\beta \wedge \omega) \\ & = \beta(\sharp \alpha) \omega \\ &= g^{-1}(\beta, \alpha) \omega \end{align} where $\beta \wedge \omega$ vanishes because it is a top form and I use $g^{-1}$ for the induced inner product on 1-forms. It also works nicely in coordinates, since if I introduce the $(n-1)$-forms $\omega_a = \iota_{\frac{\partial}{\partial x^a}} \omega$, then \begin{equation} \ast X = \iota_X \omega = X^a \iota_{\frac{\partial}{\partial x^a}} \omega = X^a \omega_a \end{equation}
My question is whether this actually generalizes to higher dimensions (I am thinking of an isomorphism $\Gamma(\Lambda^p TM) \to \Omega^{n-p}(M)$ determined by a choice of volume form) and whether it might be useful to think of the bit of Hodge star determined only by a volume form in contexts in which a metric might not be readily available.