I am trying to understand the relationship between the wedge product and linear subspace. Let $e_1,\cdots, e_4$ be the standard basis of $\mathbb{R}^4$. The wedge product $(e_1+2e_2)\wedge (3e_1+e_3+e_4) $ can be thought of as the oriented linear subspace generated by the vectors $e_1+2e_2$ and $3e_1+e_3+e_4$.
Upon expanding we get $e_{13}+e_{14}-6e_{12}+2e_{23}+2e_{24},$ where $e_{ij}=e_i\wedge e_j$. Each $e_{ij}$ can be thought of as the linear subspace spanned by $e_i,e_j$. But what role do the coefficients $1,1,-6,2,2$ play?