For some reason I have never actually thought about this too hard, but now I am getting myself confused. Suppose $M$ is a smooth manifold, and $\omega\in \Omega^k(M)$. Let $(U,\phi)$ be a coordinate chart with coordinates $(x^1,\dots, x^n)$. Then, on $U$ we can "write $\omega$ in coordinates" as: \begin{align*} \omega|_U=\sum_{i_1<\cdots<i_k}\omega_{i_1\cdots i_k}dx^{i_1}\wedge \cdots \wedge dx^{i_k} \end{align*} where the $\omega_{i_1\cdots i_k}$ are smooth functions $U$.
In this sense, the wedge products $dx^{i_1}\wedge \cdots \wedge dx^{i_k}$'s form a local coordinate frame for the bundle $\Lambda^k(T^*M)$, however, in practice when I write a differential form in coordinates, it seems that I am actually writing forms on $\Lambda^k(T^*\phi(U))$, which I guess is the point, but I am getting myself confused a bit.
For example, suppose $M=S^2$ equipped with the round metric, then I can write the volume for $\omega_g$ in the angle coordinates as: \begin{align*} \omega_g|_U=\sin\theta d\theta\wedge d\phi \end{align*} but this to me seems like a differential form on $\phi(U)$, where $\phi$ is the chart defined implicitly by its inverse: \begin{align*} \phi^{-1}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta) \end{align*} so I really think we should be writing: \begin{align*} \phi^{-1*}\omega_g=\sin\theta d\theta\wedge d\phi \end{align*} and that: \begin{align*} \omega_g|_U=\phi^*(\sin\theta d\theta\wedge d\phi) \end{align*} In general, when writing a differential form in coordinates, I think we should really be writing: \begin{align*} \phi^{-1*}\omega=\sum_{i_1<\cdots <i_k}\omega_{i_1\cdots i_k}dx^{i_1}\wedge \cdots \wedge dx^{i_k} \end{align*} and: \begin{align*} \omega|_U=\phi^*\left(\sum_{i_1<\cdots <i_k}\omega_{i_1\cdots i_k}dx^{i_1}\wedge \cdots \wedge dx^{i_k}\right) \end{align*} where the $\omega_{i_1\cdots i_k}$ are smooth function on $\phi(U)$.
Am I missing something here?