Consider two differential one forms:
$$\omega=\sum_{i=1}^N \omega_i dx^i$$ $$\omega'=\sum_{i=1}^N \omega'_i dx'^i$$
As I recall from my analysis courses, the symbols $dx$ are a particular notation for specific dual basis. It now makes sense to define the equality $\omega=\omega'$ if and only if
$$<\omega,v>=<\omega', Mv> \forall v \in \mathbb{R}^N$$
Where $M$ is the matrix that changes from the canonical basis to the basis given by the isomorphism $v^i_k = dx'^i_k$, $k=1,...,N$ (this is all speculation since we haven't covered change of coordinates with one-forms, so please correct me if I'm wrong). Is it now possible to have more general change of coordinates, namely non-linear ones? How do one forms transform under those?
EDIT: For those of you who don't dislike physics, I need this for a proof on canonical transformations of the Hamiltonians. Namely, we are supposing we have two one forms which, by hypothesis, are claimed to be equal up to an exact differential.
$$pdq-H(p,q,t)dt=PdQ-K(P,Q,t)dt+dF(P,Q,t)$$
I had a hard time interpreting the equality, since I didn't know how to interpret the $dQ$ symbol (better: I had an idea, that is taking the differential of the change of coordinates and substituting, this post is mainly for confirmation and for learning something a bit more general).