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Let $M$ be a $n$-dimensional manifold and $\alpha$ a smooth $r$-form in $M$. It is clear that $\alpha$ can be written locally as $\alpha=\alpha_1\wedge \ldots \wedge \alpha_r$ for some $1$-forms $\alpha_1,\ldots,\alpha_r$ just by considering its expression in a local chart of $M$.

In general, can we have this descomposition $\alpha=\alpha_1\wedge \ldots \wedge \alpha_r$ in the whole manifold? Maybe with the use of bump functions?

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    $\begingroup$ In Euclidean $\mathbb{R}^3$, how do we write $dx_1 \wedge dx_2 + dx_2 \wedge dx_3 + dx_3 \wedge dx_1$ as $\alpha_1 \wedge \alpha_2$? $\endgroup$
    – aschepler
    Aug 16, 2022 at 14:27
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    $\begingroup$ @aschepler $(dx_1 - dx_3)\wedge(dx_2 - dx_1)$ $\endgroup$ Aug 16, 2022 at 15:17

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In looking for the easiest counter example, my initial answer was incorrect. However, it is still not true, even locally. For example, if we take $dx\wedge dy+dz\wedge dw$ on $\mathbb{R}^4$, you cannot write this as a wedge of $1$-forms, but trying to is probably a good exercise.

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