Let $M$ be an oriented Riemannian manifold of dimension $n$. For any $\omega \in \Omega^k(M)$, we define the Hodge star operator $\star$ of a $\omega$ as the unique $n-k$ form $\star\omega$ that satisfies $$\omega \wedge (\star\omega) = \langle \omega, \omega \rangle dVol.$$
How is this operation extended to forms that now take values in a vector space $V$?
Similar to this post, I am interested in this because of its applications to Yang-Mills theory where we have $F \wedge \star F$ and where $F$ is a 2-form with values in a Lie algebra $\mathfrak{g}$. The accepted answer explains how to wedge two vector-valued forms, but I am interested in the Hodge star of two vector-valued forms.
Can this only be done locally in the sense that if $V$ has dimension $m$, then $$\omega = \sum_{i=1}^m \omega_i \otimes e_i$$ then $$\star\omega = \sum_{i=1}^m (\star\omega_i) \otimes e_i$$ where $\omega_i$ is a real-valued $k$-form and $\{e_i\}$ is a basis for $V$?
