Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an $L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential forms by

$$ \langle \alpha, \beta \rangle_g = \int_M \alpha \wedge \ast_g \beta, $$

where $\ast_g$ denotes the Hodge-star operator relative to $g$, and $\alpha, \beta$ are forms of the same degree.

**Question**: Does every inner product on $\bigwedge^\ast(M)$ as a graded
vector space come from some metric $g$? How about inner products on
$k$-forms $\bigwedge^k(M)$ for a single $k$, especially $0 < k < \dim
M$?