Why is this wedge product equal to zero?

Question

I don't understand why $de^{1}\wedge e^{3}-e^{1}\wedge de^{3}=0$ (d is the exterior derivative)

Could somebody please explain this to me? Thank you!

Answer 1

I think this follows from the following formula:

If $\alpha$ is a k-form and $\beta$ is a form, then

$$d(\alpha \land \beta) = d \alpha \land \beta + (-1)^k \alpha \land d \beta$$