There wasn't enough room in the title to explain completely: $M$ is an oriented Riemannian manifold with boundary $N$. $\alpha$ and $\beta$ are differential forms on $M$, $*$ denotes the Hodge star, and $i:N \hookrightarrow M$ is the inclusion.
Now, if I know $i^{*}\alpha$ and $i^{*}(* \alpha)$, that should be enough to uniquely reconstruct $\alpha$ on $N$. So there is hope that there is a formula for $i^{*}(*(\alpha \wedge \beta))$ in terms of $i^{*}(\alpha)$, $i^{*}(\beta)$, $i^{*}(*\alpha)$, $i^{*}(*\beta)$, and the wedge product/exterior derivative/hodge star on $N$. Is there?
My motivation is, given a connection $A$ on $M$ for a trivial $G$ bundle, trying to reconstruct $i^{*}(F_A)$ using only data that depends linearly on $A$ (this is a $\mathfrak{g}-$valued one-form but similar logic should hold).