I am pretty inexperienced in differential geometry and wanted to go through the proof of Stokes theorem. The usual way I see it formulated is for an oriented manifold with boundary $M$. There was a particular way I saw it presented before, and was wondering if there was any reference which did things in this way.
I know what I am going to say is just a translation of the usual formulation of Stokes theorem, so if there is no such reference, I will just read a more standard treatment and translate the proof into a way I like.
So $M$ is an $n$-dimensional smooth manifold, and $\Omega^k(M)$ is the space of differential $k$-forms on $M$. We have a cochain complex
$$\cdots \rightarrow \Omega^0(M) \rightarrow \Omega^1(M) \rightarrow \Omega^2(M) \rightarrow \cdots $$
where $d: \Omega^i(M) \rightarrow \Omega^{i+1}(M)$ is the exterior derivative. For each $k$, we also consider all smooth functions $[0,1]^k \rightarrow M$ (meaning smooth on the interior $(0,1)^k$, with some nice boundary conditions), and let $C_k$ be the free vector space with all those smooth functions as a basis. There are natural (?) boundary maps $\partial: C_i \rightarrow C_{i-1}$ which give us a chain complex
$$\cdots \rightarrow C_2 \rightarrow C_1 \rightarrow C_0 \rightarrow \cdots$$
If $f: [0,1]^k \rightarrow M$ is a basis element of $C_k$, and $\omega \in \Omega^k(M)$, then the pullback $f^{\ast} \omega$ is a differential $k$-form on the manifold $(0,1)^k$, and one defines a pairing $C_k \times \Omega^k(M) \rightarrow \mathbb R$ by
$$\langle f, \omega \rangle = \int\limits_f \omega = \int\limits_{(0,1)^k} f^{\ast} \omega$$
where the right hand side is the usual integration of a top form in $\mathbb R^k$. One extends $\langle f, \omega \rangle$ by linearity to all $f \in C_k$.
The version of Stokes theorem I heard is the following:
Theorem: Let $\omega \in \Omega^{k-1}$, and $f \in C_k$. Then
$$\langle f, d \omega \rangle = \langle \partial f, \omega \rangle$$
or in other words
$$\int\limits_f d \omega = \int\limits_{\partial f} \omega$$
Is there a nice reference which explains Stokes theorem in roughly the way I'm describing? If not, how is what I am saying related to the usual formulation? For one thing, I don't understand why Stokes theorem is stated for an oriented manifold. The formulation I have described says nothing about an orientation on $M$.