Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a chart $(U,x)$ - extend $x(U)$ to $\mathbb{R}^n$ and take the components).
We have the inner product of smooth $k$-forms $\langle \omega , \eta \rangle = \int \omega \wedge \star\eta$, which we can extend to complex-valued forms. Moreover every $f \in C_c^\infty(\mathbb{R}^n;\mathbb{C}^m)$ extends by zero to a complex valued $k$-form on $M$ and we can transfer $\langle \cdot , \cdot \rangle$ to $C_c^\infty$. On $C_c^\infty$ we have another inner product $\langle \varphi,\psi \rangle_{L^2} = \frac{1}{(2\pi)^n} \int \varphi \cdot \bar{\psi}$
I have problems understanding the fact that there is a smooth matrix $A$, such that $\langle \varphi,\psi \rangle = \langle \varphi , A \psi \rangle_{L^2}$ on $C_c^\infty$. Can someone bring me on the right track to reduce that to the pointwise case? I would appreciate any hints!