I am considering the following form of Stokes's theorem:
Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary $\partial M$ of $M$ with the induced orientation. Then $$\int_{M}d\omega = \int_{\partial M}\omega.$$
In Green's theorem we have a positively oriented, piecewise smooth, simple closed curve $C$ in a plane, and $D$ be the region bounded by $C$. So $C$ plays the role of $\partial M$ and $D$ of $M$.
On the other hand in Stokes's theorem the boundary $\partial M$ is a smooth manifold of dimension $n-1$.
My question is the following: is there a way to deduce Green's theorem from Stokes's theorem?