I am currently practising the wedge product, but I don't quite understand the structer overall. There is a task in my textbook marked "easy". Could anyone help me with this? I think an example would help me a lot.
Let $V$ be a real vector space, $\dim V=3, \ \ \sigma_1,\sigma_2,\sigma_3$ a basis of $V^*$, $\omega=\sum a_i \sigma_i, \ \ \eta=\sum b_i \sigma_i$ two random elements of $V^*$.Calculate $\omega \wedge \eta$ and give reasons why the wedge product is a generalization of the cross product.
Let $v,w \in V$. I know that then: $$(\omega \wedge \eta)(v,w)=\omega(v)\eta(w)-\omega(w)\eta(v)$$ But where is the connection to the cross product?