Deriving a boundary flux version of Stokes' theorem

Question

Just like how there is a flux version of Green's theorem, I want to derive a version of stokes theorem where you integrate the flux of the field perpendicular to the boundary of S. Intuitively it should be related to some kind of projection of the divergence of F over the surface. Given the usual setup of stokes, with surface S, vector field F, tangent vector to the boundary T, and normal vector n. The vector $\vec{P}$ is the vector perpendicular to $\partial S$ and parallel to S. It is given by $$\vec{T}\times\vec{n}=\vec{P}\\$$ This vector points outwards according to RHR. I want to relate the following line integral to some type of divergence of F over S. $$\oint_{\partial S}^{}\vec{F}\cdot\vec{P}ds\\$$ $$=\oint_{\partial S}\vec{F}\cdot(\vec{T}\times\vec{n})ds\\$$ $$=\oint_{\partial S}(\vec{n}\times\vec{F})\cdot\vec{T}ds\\$$ $$=\oint_{\partial S}(\vec{n}\times\vec{F})\cdot d\vec{r}\\$$ $$=\iint_S (\nabla\times(\vec{n}\times\vec{F}))\cdot\vec{n} d\sigma\\$$ I tried using an identity involving the curl of a cross product, but I wasn't able to get anywhere with that. Where do I go from here?