Let $A$ open in $\mathbb{R}^{n}$, $u=\sum_{i=1}^{n}a_{i}e_{i}$ and $v=\sum_{i=1}^{n}b_{i}e_{i}$ vectors in $\mathbb{R}^{n}$, where $\{e_{1},\dots,e_{n}\}$ denote the usual basis for $\mathbb{R}^{n}$. How can I write the general expression for a 2-form $\omega(p,u,v)$ in $\mathbb{R}^{n}$?
I know that $$\omega\left(p,\sum_{i=1}^{n}a_{i}e_{i},\sum_{i=1}^{n}b_{i}e_{i}\right)=\sum_{i,j=1}^{n}a_{i}b_{j}\omega\left(p,e_{i},e_{j}\right).$$ But $\omega(p,e_{i},e_{i})=0$ and $\omega(p,e_{i},e_{j})=-\omega(p,e_{j},e_{j})$. I'm looking for a way to write this as a combination of determinants. Can I?