I need to take the wedge product
$$w_{1k}\wedge w_{2k} = \left(\sum_I a_{Ik}d_I\right)\wedge \left(\sum_J b_{Jk}d_J\right)$$
indexed in $I$ and $J$ which are finite sets.
Can I express this as simply
$$\sum_{I,J} (a_{Ik}\wedge b_{Jk})d_I\wedge d_J$$
or something like that?
I need this because I want to prove that when there are $w_{1k}$ such that $w_{1k}\to w_1$ and $w_{2k}$ such that $w_{2k}\to w_2$, then the product $w_{1k}\wedge w_{2k}\to w_1\wedge w_2$ .
I'm considerng $w_1 = \sum_I a_I d_I$ and $w_2 = \sum_I b_I d_I$, therefore I need to know how $w_1\wedge w_2$ looks like so I can prove that $w_{1k}\wedge w_{2k}\to w_{1k}\wedge w_{2k}$