Expression for a k-form $\omega$ written locally
Page 303 John Lee Introduction to Smooth Manifolds First Edition:
In any smooth chart, a k-form $\omega$ can be written locally as:
$\omega = \Sigma'_I \omega_I dx^{i^1} \wedge .... \wedge dx^{i^k}$
Where $\Sigma'_I$ is taken over all the increasing multi-index's [edit] of length k.
Where the coefficients $\omega_I$ are continuous functions defined on the coordinate domain.
I was just wondering if somebody could explain to me why it is necessary to sum over all increasing multi-index's [edit] of length k. I think the reason that I am confused is that an earlier result showed that $\{\epsilon_I$ | $I$ is an increasing multi-index of length k $\}$ is a basis for the vector space of alternating co-variant k-tensors.
Is it strange that the summation defining a k-form locally goes over ALL the increasing multi-index's [edit] of length k? What is the connection here between the basis for the alternating co-variant k-vectors and the index of summation in expressing a k-form locally? I'd appreciate any insight in general, thanks!