Differential Forms: Writing in Terms of Local Coordinates

[Nusc]

Homework Statement

Let $$x_1,...,x_n: M \rightarrow R$$ be functions on a manifold which form a local coordinate system on some region. Show that every differential form on this region can be written uniquely in the form

$$w^k = \sum_{i_1<...<i_k} a_{i_1,...i_k}(\bf{x})dx_{1_i} \wedge .. \wedge dx_{1_k}$$

Any ideas?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Suppose you are in 3 dimensions. How would you write $d_{x_1}d_{x_3}$ in that form?