Following is an excerpt from Jeffrey M. Lee - Manifolds and Differential Geometry. On page 538 it states if $e_i$ is a frame field of a vector bundle $E\rightarrow M$ on $U\subset M$, then locally, for $\omega \in \Omega^k(M,E)$ we may write $$ \omega = \sum e_i \otimes \omega^i, $$ where $\omega^i \in \Omega^k(M)$.
My assertion is that $\omega^i$ are only defined on $U$ i.e. $\omega^i \in \Omega^k(U) $. General element of $\Omega^k(M,E)$ is of the form $ \sum \alpha_j \otimes\beta^j$ for $\alpha_j \in \Gamma(E)$ and $\beta_j \in \Omega^k(M)$. So far everything is globally defined. Now rewriting $\alpha_j$ in terms of local frame: $$ \alpha_j = \sum_i f^i_j e_i,$$ where $f^i_j$ smooth functions on $U$. Bringing everything together to one side $$ \sum \alpha_j \otimes\beta^j = \sum_{i,j} e_i \otimes f^i_j\beta^j. $$ Setting $\omega^i:= \sum f^i_j\beta_j$ has the form as in the book, but it is only well defined on $U$.
How do I arrive to conclusion?