I wasn't sure if to ask this question on math SE or physics SE, but I figured it was entirely mathematical in nature with only applications in physics.
That being said, the "metric tensor" $g$ is really a section of the bilinear form bundle $T^*M\otimes T^*M\xrightarrow{\pi}M$ such that $g_p:T_pM\times T_pM\to\mathbb{R}$ is symmetric and non-degenerate for all $p\in M$.
On the other hand, a "differential $2$-form" $\omega$ is also really a section, and $\omega_p$ also eats two vectors and outputs a scalar. Furthermore, we can integrate this $$\iint_\Omega \omega$$ over a region $\Omega\subseteq M$ of dimension $2$.
My question is: Is the metric tensor also a differential form and can we integrate it over a region of a pseudo-Riemannian manifold?
$$\iint_\Omega g$$
Would the result be anything physical or meaningful?