I read Rudin's "Principles of Mathematical Analysis". In the part of Differential Forms, he defined them formally. I particularly enjoyed the formal viewpoint, since everywhere else it seems that the fact that a form is a multi-linear function serves only to demonstrate the algebraic properties of operating with forms, thus polluting the content with useless information. But those properties come out easily under the viewpoint of Rudin's book. The problem is: I don't see how to generalize the integral that he takes over a surface in $\mathbb{R^n}$ to a manifold, since you need explicitly the jacobian from the parametrization of the surface on his definition (I don't want to appeal to Whitney's Theorem).
More explicitly, he defines a differential form to be:
$$w:=\sum \displaystyle a_{i_1,i_2,...,i_k}(x)dx_{i_1}\wedge dx_{i_2} ...\wedge dx_{i_k}$$
everything formal, and the integral of $w$ over a parametrized surface $\phi$ (with domain of parametrization $D$) is:
$$\displaystyle \int _{\phi}w:= \int _{D}\sum \displaystyle a_{i_1,i_2,...,i_k}(\phi(u))\frac{\partial(\phi_{i_1},\phi_{i_2},...,\phi_{i_k})}{\partial(u_1,...,u_k)}d\textbf{u}$$
My problem is: the jacobian in question depends on the canonical basis of $\mathbb{R}^n$. Is there any way I can preserve this viewpoint, but being able to talk meanigfully about integrals on manifolds?