The divergence theorem can be stated as
$$\bigcirc \hspace{-1.3em} \int \hspace{-.8em} \int\limits_{\partial\Omega} dA\,n_i = \iiint\limits_\Omega dV\partial_i$$
applied to an arbitrary function (usually a vector valued field) where $\partial\Omega$ is the closed surface of the volume $\Omega\subset\mathbb{R}^3$ and $n_i$ is the $i$th component of the surface normal vector $\vec n$.
Is there a similar correspondence between e.g.
$$\bigcirc \hspace{-1.3em} \int \hspace{-.8em} \int\limits_{\partial\Omega} dA\,n_i n_j$$
and another volume integral?