Let $A$ be the four-potential, then we know that we can form the electromagnetic tensor as $F=dA$. This is usually done as a way to have a better writing of Maxwell's equations. So, to simplify the equations and make then covariant we simply notice that we can join electric and magnetic potentials in a single one and then take the derivative and then we find Maxwell's equations from $d^2A = 0$ and all of that.

That's all fine, but $F$ is a differential $2$-form and such objects are highly geometrical. What is then the interpretation of the electromagnetic tensor? I know that $2$-forms represents "objects that perform $2$d measures", but in this case what does $F$ measures?

Until now all approaches I've seem to introduce this tensor were mainly to rewrite something in a better way.