Inner product on pseudoriemannian manifold

Question

In this topic: Scalar product on manifold. Henry said $(\omega, \eta)=\int_M \omega \wedge \star \eta=\int_M \left<\omega, \eta \right>\mathrm{d vol}$ is an inner product on forms, where $\left<\cdot, \cdot \right>$ is the pointwise inner product induced by riemannian metric i.e. $\left<\alpha, \beta \right>:=g^{ij}v_iw_j$ for $1$-forms $\alpha=v_i e^i$ and $\beta=w_i e^i$, $\left<\alpha_1 \wedge \ldots \wedge \alpha _p, \beta_1 \wedge \ldots \wedge \beta_p \right>:=\det(\left<\alpha_i, \beta_j \right>)$ etc. I assume $(\omega, \eta):=\int_M \left< \omega, \eta\right> \mathrm{dvol}$ is not an inner product on pseudoriemannian manifold, since $\left<\alpha, \beta \right>:=g^{ij}v_iw_j$ is no longer an inner product. Can we save this structure somehow? Does exist an inner product on differential forms on pseudoriemannian manifolds? ;>