Cartan lemma

Let $V$ be a vector space (typically $V = T_{p}^{*} M$ ). Given linearly independent elements $v_{1}, \dots, v_{p} \in V$ and provided that $w_{1}, \dots, w_{p}$ are such that

v_{1} \land w_{1} + \dots + v_{p} \land w_{p} = 0

in $Λ V$ then there exist scalars $h_{i j} = h_{j i}$ such that

w_{i} = \sum_{j = 1}^{p} h_{i j} v_{j}

Moreover, $h_{i j} = h_{j i}$ .

Proof. See @spivak1999comprehensive vol 3 page 18. $◼$

Related: Do not confuse with Cartan formula.