Below I am trying to express my understanding of complex differentials. My question, then, is
"Is my summary below correct, with respect to what I discuss there?"
Given a manifold $M$, a differential 1-form $\omega$ is a smooth map on $TM$ whose restriction on each $T_p M$ is a linear functional. If given a chart with coordinate functions $x^i$, at each point $p$ the derivations $X^i(p)$ form a basis, and the functionals defined by $dx_j(X^i(p)) = \delta^i_j$ form a basis for the linear functionals. As such, any differential form can be written as a linear combination of the $dx_j$'s with coefficients that are differential functions in $p$. That is, for any $X \in T_pM$ we get $$\omega(p)(X) = \sum_{j = 1}^n f^j(p) dx_j(X).$$
The integral of $\omega$ against a path $\gamma : [0, 1] \to M$ is then $$\int_\gamma \omega := \int_0^1 \omega(\gamma(t))(\gamma^\prime(t)) dt.$$
For a Riemann surface, a coordinate $z$ can be written as $z = x + i y$, and the complex differential $dz$ can then be written as a complex linear combination in $dx$ and $dy$. Everything else works as expected. A complex differential $\omega$ is said to be holomorphic, meromorphic, etc. if it can be written in every coordinate system as $f(z) dz$ with $f$ being holomorphic, meromorphic, w/e.
I understand the construction of the wedge product, but it does not make sense. I'd appreciate anyone who can help me make sense of this?