Conceptual doubt about pullback and local coordinates

I'm studying differential forms in manifolds through the book: Differential Geometry and Topology With a View to Dynamical System, by Keith Burns.

I have a conceptual question and I would like you to answer me, if possible. (I'm studying Analysis in $\mathbb{R}^n$ and this part of differential forms I have to follow with a book on differential geometry.)

Let $M$ be a smooth manifold and $\psi : U \subseteq \mathbb{R}^n \longrightarrow V \subseteq M$ be chart. If $\omega$ is a differential form in $M$, then I'm wrong to say that $\omega$ can be written in local coordinates as$$\omega = \sum_{j_1 < \dots < j_k} f(x)\,dx_{j_1} \wedge \cdots \wedge dx_{j_k} $$is the same thing to calculate $\psi^*\omega?$