I am a physicist and consequently I do not have the same rigor as a mathematician but I'm interested in better understanding the concept of differential form.
I have seen that the p-forms are completely antisymmetric covariant tensors (0, p) that allow us to define the integral operation on generic spaces in a rigorous way.
A generic p-form is written in terms of the basis elements via the wedge product: $$\omega = \omega_{\mu_1...\mu_p}dx^{\mu_1}\wedge...\wedge dx^{\mu_p}$$ and a p-dimensional integral can be seen as a map between a p-form and real numbers: $$\int: \omega \to \mathbb{R}$$ I also understood that seeing the integral in this way is better because the wedge product is a generalization of the vector product on generic spaces and therefore allows us to define the infinitesimal volume on these spaces.
My doubt is the following: if we consider a simple surface integral in an Euclidean space 2D like: $$\int_S f(x,y) dxdy$$ we have to interpret $dxdy$ like a wedge product $dx \wedge dy$ that is antisymmetric. So why in the courses of analysis these integrals are made by freely exchanging the two one-forms without taking into account the minus sign?
Briefly, while mathematicians speak of integrating $p$-forms over $p$-dimensional domains, in practice we often integrate densities, i.e., absolute values of $p$-forms. That is, we ignore (and often forget) the order of differentials. "Scalar line integrals," "scalar surface integrals," and volume integrals of multivariable calculus are examples. If we were more careful in this regard, we'd write, for example, $$ \int_{S} f(x, y)\, |dx \wedge dy|,\qquad \int_{D} \rho(x, y, z)\, |dx \wedge dy \wedge dz|, $$ or explicitly note somewhere that $dx\, dy = |dx \wedge dy|$, $dx\, dy\, dz = |dx \wedge dy \wedge dz|$, etc.
Instances where we integrate differential forms proper include