I am studying gauge theory, and we derive the Lagrangian for electrodynamics by wanting the Lagrangian to be gauge invariant under U(1) symmetry group. That is, invariant under the phase rotation, $$\psi(x) \rightarrow e^{i\alpha(x)} \psi(x).$$ We derived the covariant derivative, $$D_{\mu} = \partial_{\mu} + A_{\mu},$$ where $A_\mu$ is the gauge field, and transforms as $$A_\mu \rightarrow A_\mu -\partial_\mu \alpha(x).$$ We then found the electromagnetic field tensor by saying that $[D_\mu, D_\nu]$ must be gauge invariant also. When we substitute $D_\mu = \partial_\mu + A_\mu$ into $[D_\mu, D_\nu] \psi(x)$, we can derive, $$[D_\mu, D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu.$$ This is in fact the electromagnetic field tensor, $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$ How does this derivation show that the electromagnetic field tensor therefore has geometrical origins, and is fundamentally geometric in nature? Is this because the commutator of the covariant derivative is the comparison of parallel transporting an object in one direction, and then the opposite direction? Could there be a pictorial visualization of what is going on here somehow? Would the $F_{\mu \nu}$ being geometric in this way account for why it is called a curvature 2-form, since it is related to the curvature of the manifold it is defined over? I want to gain an intuitive understanding of what is happening here, so that the link to the geometry is clear.

I also wanted to ask if I am missing some factors of i here? I am unsure if a few of my derivations are wrong or not.