Let M be a smooth manifold. Can every k-form $\omega$ on M be written as a sum of k-forms, that are wedge products of 1-forms, i.e. $\omega = \sum\_{i=0}^n \alpha\_1^{(i)} \wedge \ldots \wedge \alpha\_k^{(i)} $, where $\alpha\_l^{(i)} \in \Omega^1(M) $ ? If M is compact, one can cover M be finitely many charts and use a partition of unity to see that this holds...but how about the general case? 



k-form: sum of wedge products of 1-forms?
The answer is yes and you need transversality in some form. Here we use Whitneys embedding theorem, but a weaker statement would suffice. Embed $M \subset \mathbb{R}^m$. Thus there is a monomorphism $TM \to M \times \mathbb{R}^m$ of vector bundles, dualizing to an epimorphism $M \times \mathbb{R}^m \to T^{\ast} M$. Let $a\_1,\ldots ,a\_m$ be the images of the basis vectors. You can write any $k$-form on $M$ as a linear combination with $C^{\infty} (M)$-coefficients of the forms $a\_{i\_1} \wedge \ldots a\_{i\_k}$. Done.


