Consider the Generalized Stokes Theorem:
\begin{equation} \int_Md\omega = \int_{\partial{M}} \omega \end{equation} Here, $\omega$ is a k-form defined on $R^n$, and $d\omega$ (a k+1 form defined on $R^n$) is the exterior derivative of $\omega$. Let M be a smooth k+1-manifold in $R^n$ and $\partial{M}$ (the boundary of M) be a smooth k manifold.
I know that the above theorem is simply a generalization of well-known vector calculus theorems. However, I am looking for the intuition behind the Generalized Stokes Theorem itself.
I started off by defining the exterior derivative at a point p in $R^n$ as: \begin{equation} d\omega_p =\lim_{|vol|\to 0}\frac{\int_{\partial{vol}} \omega}{|vol|} \end{equation}
In this case, "$vol$" represents a k+1 "parallelpiped" in $R^n$ that contains point p (with $|vol|$ being its "volume"). $\partial{vol}$ represents the boundary of this k+1 "parallelpiped", a k "parallelpiped" itself.
With this definition (assuming it is correct), can we say that $\omega$ represents an infinitesimal "flux" element through $\partial{vol}$ which would imply that $d\omega_p$ is simply the "flux density" at a point p?
If the above is true, can we take the idea that (when applying the Generalized Stokes Theorem) the interior "fluxes" through each $\partial{vol}$ within M cancel out leaving us with the total "flux" out of $\partial{M}$ as the intuition behind the Generalized Stokes Theorem?
Any help is much appreciated.
Yes, this is very good intuition for the theorem. All of this can be made precise.
In fact, it is made precise on pages 188-190 of the second edition of Arnold's Mathematical Methods of Classical Mechanics. There Arnold gives the following theorem, where $\omega$ is a given $k$-form on an $n$-dimensional manifold $M$.
First, the geometric setup. The idea is to construct a $(k+1)$-form by computing its value on a given list of $k+1$ vectors.
So let $\xi_1,\ldots,\xi_{k+1}$ be tangent vectors in $T_xM$, where $x\in M$. We can pull these back to $\mathbb{R}^n$ in the following way. Choose a coordinate system $\phi:U\to\mathbb{R}^n$ for $x\in U$ with $\phi(x)=0$. The preimages of the $\xi_i$ under the differential of $\phi^{-1}$ are tangent vectors $\xi_i^*$ in $T_0\mathbb{R}^n$. But we can naturally identify this tangent space with $\mathbb{R}^n$ itself. Let $\Pi^{*}$ be the parallelepiped in $\mathbb{R}^n$ spanned by these vectors. The map $\phi^{-1}$ carries this linear parallelepiped onto a "curvilinear parallelepiped" $\Pi$ in $M$ (this is like what you call "vol"), the boundary of which is a $k$-chain, $\partial\Pi$ (which is like what you call the boundary of "vol"). Define $F$ to be the integral of the given $k$-form $\omega$ on the boundary of this curvilinear parallelepiped:
$$F(\xi_1,\ldots,\xi_{k+1})=\int_{\partial\Pi}\omega$$
Theorem. There is a unique $(k+1)$-form $\Omega$ on $T_xM$ which is the principal $(k+1)$-linear part at zero of $F$, i.e.
$$F(\epsilon\xi_1,\ldots,\epsilon\xi_{k+1})=\epsilon^{k+1}\Omega(\xi_1,\ldots,\xi_{k+1})+o(e^{k+1})$$
as $\epsilon\to0$. The form $\Omega$ does not depend on the choice of coordinates. And this unique form $\Omega$ is precisely $d\omega$ (in the usual calculation).
In this sense, $d\omega$ is indeed a kind of "flux density" of $\omega$.
The proof of the generalized Stokes theorem then follows, as littleO suggests in a comment, by making precise the idea of chopping up the manifold into little parallelepipeds. One just keeps track of all the orientations as one integrates the flux density over these little regions, $\int_{M}d\omega$. When you do so carefully, the interior fluxes cancel, leaving only the flux along the boundary, $\int_{\partial M}\omega$.