Is there a similar equivalence like the divergence theorem for surface integrals non-linear in the normal vector?

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?

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@Matt thanks, I tried \oiint but didn't think \oint would work instead... –  Tobias Kienzler Sep 4 '12 at 12:26
That's ok, I was just curious and the result turned out ok. Somebody probably knows how to center the limit properly as well. –  Matthew Pressland Sep 4 '12 at 12:28
@MattPressland I found a nice \oiint here –  Tobias Kienzler Sep 5 '12 at 13:18
That looks better! (I didn't actually realise the circle was supposed to go through both integrals). –  Matthew Pressland Sep 5 '12 at 13:29