If we have a surface having a position vector parameterized $R$ by two variables $u,v$, we can find it's tangent vectors $\partial_u R ,\partial_v R$ by taking derivative of the position vector with each variable. After finding the tangent vectors, we can correspond them to the tangent one forms $r_u$ and $r_v$. Using this, we can find the normal two form as:
$$ n= \frac{r_u \wedge r_v}{|r_u \wedge r_v| }= n_3 dx \wedge dy + n_1 dy \wedge dz + n_2 dz \wedge dx$$
Now suppose we integrate the above across the surface and used stokes:
$$ \int_{\partial \omega} n = \int_{\omega} dn$$
Does applying stokes here even make sense? Because I derived $n$ from the tangent forms, hence it was a thing which existed on the surface, strictly speaking, the components of $n$ is defined only on the surface itself. What theorem tells us that it extends, and is the extension unique?