I have seen several proofs of Stokes theorem; all of them involves heavy calculation and simplification of partial derivatives and computing cross products term by term.
The theorem is, however, extremely easy to understand intuitively. The total "circulation" around a curve $\gamma$ equals the sum of all "microscopic circulation" on a surface spanned by $\gamma$. For a diagram of "microscopic circulation", see here.
I hope I can find a proof which illustrates directly how the idea of summing over microscopic circulation. Algebraic simplification of partial derivatives doesn't seem to show this.