In mathematics and physics, especially gauge theory, there are many different but related notions of wedge products when discussing vector space- and vector bundle-valued differential forms. For example, if $U,V,W$ are finite-dimensional $\mathbb{R}$-vector spaces and $\mu:U\times V\to W$ is a bilinear map, then we always can define a wedge product of the type
$$\wedge_{\mu}:\Omega^{k}(\mathcal{M},U)\times\Omega^{l}(\mathcal{M},V)\to\Omega^{k+l}(\mathcal{M},W)$$
via the usual detinition, i.e.
$$\alpha\wedge_{\mu}\beta:=\sum_{i=1}^{\mathrm{dim}_{\mathbb{R}}(U)}\sum_{j=1}^{\mathrm{dim}_{\mathbb{R}}(V)}(\alpha^{i}\wedge\beta^{j})\mu(e_{i},f_{j})$$
where $(e_{i})_{i=1}^{\mathrm{dim}_{\mathbb{R}}(V)}$ is a basis of $V$ and $(f_{j})_{j=1}^{\mathrm{dim}_{\mathbb{R}}(U)}$ is a basis of $U$ and where $\alpha^{i}\in\Omega^{k}(\mathcal{M})$ and $\beta^{j}\in\Omega^{l}(\mathcal{M})$ are the coordinate forms with respect to this basis. There are many different examples where this appears in the literature: Choosing the vector space scalar product $\mu:\mathbb{R}\times V\to V$ leads to a wedge product between real-valued and $V$-values forms, choosing an $\mathrm{Ad}$-invariant inner product on a Lie algebra $\mathfrak{g}$ leads to the "trace" of $\mathfrak{g}$-valued forms, which is for example used in the definition of Chern-Simons forms, and when taking a Lie-bracket, we recover the well-known wedge product $[\cdot\wedge\cdot]$.
Now, in many applications, one also has to deal with bundle-valued forms and in this context, there are also many different type of wedge products. I would imagine that every product in this context can be defined in the following unified framework:
Let $E,F,G$ be three smooth $\mathbb{R}$-vector bundles. Furthermore, let $\mu\in\Gamma^{\infty}(E^{\ast}\otimes F^{\ast}\otimes G)$, which we can view as a smooth assignment $\mathcal{M}\ni p\mapsto\mu_{p}$ such that $\mu_{p}:E_{p}\times F_{p}\to G_{p}$ is a bilinear map. Then we define a product of the type
$$\wedge_{\mu}:\Omega^{k}(\mathcal{M},E)\times\Omega^{l}(\mathcal{M},F)\to\Omega^{k+l}(\mathcal{M},G).$$
For this, we take a local frame $\{e_{a}\}_{a=1}^{\mathrm{rank}(E)}\subset\Gamma^{\infty}(U,E)$ of $E$ and a local frame $\{f_{b}\}_{b=1}^{\mathrm{rank}(F)}\subset\Gamma^{\infty}(V,F)$ of $F$ on some open subsets $U,V\in\mathcal{M}$. Then we can write any $\alpha\in\Omega^{k}(\mathcal{M},E)$ and every $\beta\in\Omega^{l}(\mathcal{M},F)$ as
$$\alpha\vert_{U}=\sum_{a=1}^{\mathrm{rank}(E)}\alpha^{a}e_{a}\hspace{1cm}\text{and}\hspace{1cm}\beta\vert_{V}=\sum_{b=1}^{\mathrm{rank}(F)}\beta^{b}f_{b}$$
for some local coordinate forms $\alpha^{a}\in\Omega^{k}(U)$ and $\beta^{b}\in\Omega^{l}(V)$. We then define the wedge-product $\wedge_{\mu}$ locally as
$$(\alpha\wedge_{\mu}\beta)\vert_{U\cap V}:=\sum_{a=1}^{\mathrm{rank(E)}}\sum_{b=1}^{\mathrm{rank(F)}}(\alpha^{a}\wedge\beta^{b})\mu(e_{a},f_{b}),$$
where $\mu(e_{a},f_{b})\in \Gamma^{\infty}(U\cap V,G)$ has to be understood as $\mu(e_{a},f_{b})(p):=\mu_{p}(e_{a}(p),f_{b}(p))$ for all $p\in U\cap V$. This is well-defined and independent of the choice of local frames.
A particular example of this would be a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$. When choosing the adjoint bundle $\mathrm{Ad}(P):=P\times_{\mathrm{Ad}}\mathfrak{g}$ of some principal bundle $P$, we recover for example the wedge-product used in the definition of the Yang-Mills action, when using the bundle metric on $\mathrm{Ad}(P)$ induced by an $\mathrm{Ad}$-invariant inner product on $\mathfrak{g}$.
Are such general type of wedge-products discussed in the literature and does anyone have some references for them?
Context: I am preparing some text on mathematical gauge theory and for this, I include some discussion of preliminaries including vector bundle valued differential forms and I would really like to treat them in a very general framework.
Best Answer
The most general definition I know is the following. Every fiberwise bilinear form $\eta: V_1 \times V_2 \to W$ of vector bundles $V_1, V_2, W$ over $M$ gives rise to the wedge product of vector-bundle-valued differential forms by $$(\alpha \wedge_\eta \beta)_m (X_1, \dots X_{p+q}) = \frac{1}{p!q!} \sum_{\sigma \in S(p+q)} \mathrm{sign}(\sigma) \eta_m\bigl(\alpha_m(X_\sigma(1), \dots, X_{\sigma(p)}), \beta_m(X_{\sigma(p+1)}, \dots, X_{\sigma(p+q)})\bigr)$$ where $\alpha \in \Omega^p(M; V_1), \beta \in \Omega^q(M, V_2), m \in M$ and $X_i \in T_m M$. No choice of basis or local frame needed.
A discussion of wedge products is this sense can be found e.g. in the Monastir Lecture Notes by Neeb.