Consider a smooth vector bundle $E$. I would like to define a wedge-product of the form
$$\wedge:\Omega^{k}(\mathcal{M},\mathrm{End}(E))\times\Omega^{l}(\mathcal{M},E)\to\Omega^{k+l}(\mathcal{M},E)$$
Now, I have seen two definition in the literature, but can't see why they are equivalent:
(1) Lets take some form $F\in\Omega^{k}(\mathcal{M},\mathrm{End}(E))$. Locally, we can write \begin{align*}F\vert_{U}=\sum_{i}F^{i}\otimes f_{i}\end{align*} where $F^{i}\in\Omega^{k}(U)$ are the coordinate forms and where \begin{align*}\{f_{i}\}\subset\Gamma(U,\mathrm{End}(E))\cong\mathrm{End}_{C^{\infty}(U)}(\Gamma(U,E))\end{align*} is a local frame on $U$ of the bundle $\mathrm{End}(E)$. Then, we define $F\wedge\omega$ locally as \begin{align*}(F\wedge\omega)\vert_{U}:=\sum_{i,j}(F^{i}\wedge\omega^{j})\otimes f_{i}(e_{j}),\end{align*} where $\omega\vert_{U}=\sum_{j}\omega^{j}\otimes e_{j}\in\Omega^{l}(\mathcal{M})$ is the local form of $\omega$.
(2) If $F\in\Omega^{k}(\mathcal{M},\mathrm{End}(E))$ and $s\in\Gamma(\mathcal{M},E)$, then we can define a form $$F(s)\in\Omega^{2}(\mathcal{M},E)$$ via $F(s)(X,Y):=F(X,Y)(s)$, where we use that $\Omega^{k}(\mathcal{M},\mathrm{End}(E))\cong L^{k}_{\mathrm{alt}}(\mathfrak{X}(\mathcal{M}),\Gamma(\mathcal{M},\mathrm{End}(E))$ as well as $\Gamma(\mathcal{M},\mathrm{End}(E))\cong\mathrm{End}_{C^{\infty}(\mathcal{M})}(\Gamma(\mathcal{M},E))$. Then, we define the wedge-product locally as
$$(F\wedge\omega)\vert_{U}:=\sum_{i}F(e_{i})\wedge\omega^{i}$$
where in the sum, this is now the obvious wedge-product $\wedge:\Omega^{k}(\mathcal{M},E)\times\Omega^{l}(\mathcal{M})\to\Omega^{k+l}(\mathcal{M},E)$.