Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, \dots, n$, $f_{ij}$ is a smooth function.
If for every $v\in C^1(\overline{\Omega})$ there holds $$ \int_{\partial\Omega} \sum_{i,j=1}^n f_{ij}v \nu^j = 0 $$ where $\nu$ denotes the normal vector then can we conclude that $\sum_{i,j=1}^n f_{ij}\nu^i = 0$ on $\partial\Omega$?
I tried "flatening out the boundary" near a point and the applying Du Bois-Reymond's lemma. But what I get doesn't actually fulfill all the conditions to use Du Bois-Reymond's lemma.
If someone can think of another approach that's worth trying, I appreciate the input!