The elementary tensors $xotimes cdots otimes x$, with all factors equal, are all in the subspace $vee^kscrH$. Do they span it?
Solution. Yes. Indeed, take $$eex ea &quad (x+y)otimes (x+y)-xotimes x-yotimes y\ &=xotimes y+yotimes x\ &=sqrt{2!}xvee y eea eeex$$ for example.