Showing exterior product is skew-commutative

Question

I am learning about the exterior product and wish to show the skew commutative property. Here is my work:

I am confused on how to handle the notation of $(\omega^k \wedge \omega^l)$ as at the end I want to be able to write $(\omega^l \wedge \omega^k)$ How is this handled?

Answer 1

Hint: Write $\omega^k$ in standard basis $dx^1, dx^2,\ldots, dx^n$ and note that $dx^i\wedge dx^j=-dx^j\wedge dx^i$ for $i\neq j$ according to your note. now compute $w^k\wedge \omega^\ell$.

Try yourself, and then look at Loring TU book for direct proof.