While at the vector space level, the pairing might seem slightly forced, we can derive it naturally by adding structure.
Given a vector space $V$, we have a graded commutative ring $\bigwedge V = \bigoplus_i \bigwedge^i V$.
Given $\phi\in V^*$, it naturally extends to a (graded) derivation $d_{\phi}$ of degree $-1$ on $\bigwedge V$. Since $d_{\phi}^2=0$ and $d_{\phi+\psi}=d_{\phi}+d_{\psi}$, we can extend the action of $V^*$ to an action of $\bigwedge V^*$. The pairing is just the action restricted to a single degree.
Elaboration on the constructions:
First, we need to see that specifying a derivation by how it acts on generators is actually well defined. Note that, $\bigwedge V = T(V)/(v\otimes v\mid v\in V)$ is a quotient of the tensor algebra, Given any $\phi \in V^*$, we can define a derivation $d_{\phi}$ of $T(V)$ extending $\phi$, and because every element of $T(V)$ can be written in a unique way, such a derivation is well defined. For any degree $-1$ derivation $d$ we have $d(v^2)=(dv)v-v(dv)=0$, and so $d$ vanishes on the ideal defining $\bigwedge V$, and thus passes to a well defined map there.
To see that derivations extend to an action of $\bigwedge V^*$, we have that if $d:V\to A$ is a linear map of a vector space into an algebra such that $(d(v))^2=0$ for every $v\in V$, then there exists a unique map $\bigwedge V \to A$ extending $d$. However, care must be taken here, as we want $A$ to be a graded algebra and we want $d(V)\subset A_1$.
Unfortunately, because we wish our map to take values in $\operatorname{End}_k(\bigwedge V)$, which is not commutative, we can't just use the universal property of $\bigwedge V$ being the free graded commutative algebra generated in degree $1$, and we have to* do things at the level of the tensor algebra and show that things descend.
All these are related to various structures present in differential forms and vector fields, and the interaction between them (e.g. Lie derivatives), which can be extended further to structures in Hochschild homology and cohomology. There are also analogies to be made between cup and cap products in algebraic topology.
Other related ideas worth looking into are the variants of the Schouten bracket.
Note that most of the related structures are not entirely linear, and that the structure we have here here is merely a linear approximation to them.
*No, we probably don't have to. I just can't think of a cleaner way to do it at the moment. If anybody has suggestions, please let me know.
Best Answer
Let $(e_i)_{i=1}^n$ be a basis for $E$ and $e^i$ the corresponding dual basis. In what follows I will use the natural isomorphism $F^{*} = \left( E^{*} \right)^{*} \cong E$ and Einstein's summation convention.