The first 10 pages of the following link may be helpful; it shows probably the simplest concrete nontrivial 2-form:

https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2for...

The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.

Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".