Let $M$ be a compact orientable smooth manifold with boundary. Stokes's Theorem states $\int_{\partial M} \omega = \int_M d\omega$, where $\omega$ is an $n-1$ form on $\partial M$. Via the difference map in the long exact sequence for de Rham cohomology, we have a map $H^{n-1}_{DR}(\partial M) \to H^n_{DR}(M, \partial M)$. It is a classic result that these two groups are isomorphic to $\mathbb{R}$ (assuming real coefficients).
As I understand, does Stokes's Theorem yield an isomorphism between these two groups? I apologize if this question is not posed clearly. I am trying to understand the cohomological implication of Stokes's Theorem, but the Wikipedia page states it just yields a homomorphism between de Rham cohomology and singular cohomology.