In terms of a basis \({\beta^{\mu}}\) of \({V^{*}}\), we can write a \({k}\)-form \({\varphi}\) as

\(\displaystyle \varphi=\frac{1}{k!}\sum_{\mu_{1},\dotsc,\mu_{k}}\varphi_{\mu_{1}\dots\mu_{k}}\beta^{\mu_{1}}\wedge\dotsb\wedge\beta^{\mu_{k}}.\)

Δ The above way of writing the components is not unique, and others are in common use, the main alternative omitting the factorial. |

The advantage of the expression above is that, with our isomorphism convention, the component array can be identified with the anti-symmetric covariant tensor component array in the same basis:

\(\displaystyle \varphi\mapsto\frac{1}{k!}\varphi_{\mu_{1}\dots\mu_{k}}\sum_{\pi}\textrm{sign}\left(\pi\right)\bigotimes_{i}\beta^{\pi\left(i\right)}=\varphi_{\mu_{1}\dots\mu_{k}}\beta^{\mu_{1}}\otimes\cdots\otimes\beta^{\mu_{k}}\)

Here we have dropped the summation sign in favor of the Einstein summation convention, and the last equality follows from the anti-symmetry of the component array.