In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads:
Exercise. Show that, if $e_1,\ldots,e_m$ is a basis for the ($\mathbb{C}$-vectorspace) $E$, then the images of the vectors $(e_i\wedge e_j)\otimes e_k$, for all $i
and $i\le k$, form a basis of \[ E^{(2,1)} := \left.{\textstyle\bigwedge^2E}\otimes E\middle/\left((u\wedge v)\otimes w - (w\wedge v)\otimes u - (u\wedge w)\otimes v\:\middle|\:u,v,w\in E\right)\right.. \]
First, I do not see why, for $i