First define the exterior derivative for forms defined on an open set $U \subseteq \mathbb{R^n}$. This uses the notion of integration of a $p$-form over a singular $p$-chain, which needs only the integration of $\mathcal{C}^{\infty}$-functions over compact subsets of $\mathbb{R}^p$ and runs as follows. A singular $p$-cube in $U$ is a $\mathcal{C}^{\infty}$-map $\sigma : I^p \rightarrow U$, where $I := [0,1]$ is the closed unit interval. Let $\Omega^p(U) := H^0(U;\wedge^pT^*U)$ the space of alternating $p$-forms on $U$; then each $\omega \in\Omega^p(U)$ pulls back to a top form $\sigma^*\omega$ $=$ $f dx \in \Omega^p(I^p)$ with $f \in \mathcal{C}^{\infty}(I^p)$ and $dx = dx_1 \wedge \cdots \wedge dx_p$ the canonical volume element of $\mathbb{R}^p$. It thus has an integral $$ \int_{\sigma} \omega := \int_{I^p} f dx, $$ and, in fact, this it is what differential forms are made for: born to be integrated.
Next define the vector space of $p$-chains to be the free $\mathbb{R}$-vector space on the singular $p$-cubes, so that a $p$-chain $c_p$ is a formal linear combination of singular $p$-cubes: $$ c_p = \sum_{i=1}^k \gamma^i \sigma_i \quad,\quad k\in\mathbb{N}, \gamma \in \mathbb{R}.\tag{1} $$ The integral then extends to $p$-chains by linearity; $$ \int_{c_p}\omega := \sum_{i=1}^k \gamma^i \int_{\sigma_i} \omega. $$
As a next ingredient we need that any $p$-chain $c_p$ has a boundary $\partial c_p$ which is a $(p-1)$-chain. We first define it on singular $p$-cubes $\sigma$ by $$ \partial \sigma := \sum_{j=1}^p (-1)^j (\sigma \circ d^j_- - \sigma \circ d^j_+), $$ where the singular $(p-1)$-cubes $d^j_{\mp}$ in $I^p$ define the $j$-th front and back boundary faces: $$ d^j_-(x^1, \dots, x^{p-1}) := (x^1, \dots, x^j, 0, x^{j+1}, \dots, x^{p-1}), $$ $$ d^j_+(x^1, \dots, x^{p-1}) := (x^1, \dots, x^j, 1, x^{j+1}, \dots, x^{p-1}). $$ We extend this boundary operator to $p$-chains by linearity: $$ \partial c_p := \sum_{i=1}^k \gamma^i \partial \sigma_i $$ with $c_p$ given by (1).
As a last ingredient we need that a point $P \in U$ and a $p$-tuple of vectors $X:= (X_1, \dots, X_p)$ (viewed as tangent vectors at $P$ to $U$) define a singular $p$-chain $$ [X]_P : I^p \rightarrow U $$ via $$ [X]_P(x^1, \dots, x^p) := P+\sum_{k=1}^p x^k X_k $$ as soon as the $X_k$ are so small that all the $P+x^kX_k$ are in $U$ for all $k$. In fact, these simple linear singular chains are all what is needed of this formalism to define the exterior derivative, to which we proceed next.
After this preliminaries, we now want, given $\omega \in \Omega^p(U)$, define its exterior derivative $d\omega \in \Omega^{p+1}(U)$. We do this pointwise at any point $P \in U$ by exhibiting the value $d\omega_P$, as an alternating $(p+1)$-form, takes on any $(p+1)$-tuple of (tangent) vectors $(X_1, \dots, X_{p+1})\in (\mathbb{R^n})^{p+1}$. We define $$ \fbox{$d\omega_P(X_1, \dots, X_{p+1}) := \lim_{t \rightarrow 0} \dfrac{1}{t^{p+1}} \int_{\partial([tX]_P)} \omega.$} $$
Finally, for the general case of the exterior derivative of a $p$-form $\omega$ on an $n$-dimensional manifold $M$, just take charts $\phi: V \rightarrow U$ with $V$ open in $M$, $U$ open in $\mathbb{R}^n$, with the $V$ covering $M$, and put $$ (d\omega)|V := \phi^*d\eta \quad \text{with}\quad \eta := (\phi^{-1})^*(\omega|V) \in \Omega^p(U). $$ The transformation formula for multivariate integrals then shows that the $d\omega|U$ glue well on the overlaps, thus yielding a global well-defined $d\omega$.
Loosely speaking, this defines the exterior derivative as a "volume derivative", a flux density through the boundary of an infinitesimal $(p+1)$-dimensional parallelepiped and so has as a built-in an infinitesimal version of Stokes' Theorem.