How to visually think about the difference $dw_1 - dw_2$ between two 2-forms?

Question

Let's say that I have two differential 2-forms $dw_1$ and $dw_2$ that each describe separate smooth 2-dimensional manifolds with/without boundary. Each has its own differential 1-form $w_1$ and $w_2$ that provide a connection over the underlying parameter space. Each also is characterized by their respective Euler characteristics $\chi_1$ and $\chi_2$ from the Gauss-Bonnet theorem. For instance, I can think of $dw_1$ and $dw_2$ describing two different torii, or one sphere and one torus, etc. But, what does the difference of 2-forms $dw_1-dw_2$ mean? How can I think of it intuitively? I know the connected sum allows comparison of $\chi_1+\chi_2$, but the difference? I ask only because I saw in a paper that $dw_1-dw_2$ allows a way to compare geodesic sprays between the two surfaces (i.e., they will both have the same sprays if the difference is antisymmetric), but how do I think about $dw_1-dw_2$ in general in the simplest intuitive way possible, i.e. pictorially? I doubt the picture is that removing the parts of $dw_2$ from $dw_1$, and I recently asked the following question: Insight on difference between Euler characteristics of 2 manifolds: $\chi(U)-\chi(V)$?