If we have a Lie-algebra-valued one-form $\omega:\Gamma(TP)\longrightarrow T_eG$ where $(P,\pi,M)$ is a principal $G$-bundle, how is the map "$\omega_p$", produced by evaluating the one form at a particular point $p\in P$ defined? If we evaluate a vector field at a point we get a vector, so my guess is something like $$\omega_p:T_pP\longrightarrow ???,$$ but I'm not sure what it would mean to "evaluate an element of the Lie algebra at a point", so I can't extend my guess to the target of the map. The reason I ask is because at around the 12 minute mark in this lecture https://www.youtube.com/watch?v=KhagmmNvOvQ from Frederic Schuller's "Lectures on the Geometric Anatomy of Theoretical Physics", he defines the Yang-Mills field $\omega^U:=\sigma^*\omega$ (with $\sigma$ a section of the principal bundle) as taking a vector field to an element of the Lie Algebra, but then at 22:50 of that same lecture he plugs a vector into $\omega^U$, writing $\omega^U(v)$ with $v\in T_mU$ (where $U$ is a chart domain of $M$).
I have a number of closely related confusions that I think pretty much boil down to this one, so any help would be much appreciated.