Let $A$ be a set of generators of $G=S_n$; assume $e\in A$, $A=A^{-1}$. Let $V = K^n$, $K$ a field. Consider the natural action of $G$ on $V$ (namely, $g(e_i) = e_{g(i)}$) and on $W = V\wedge V$ (namely, $g(e_i \wedge e_j) = e_{g(i)}\wedge e_{g(j)}$). Must there be a $w\in W$ such that the vector space spanned by $A^k w$ equals $W$ for some $k = o(n^2)$? What about for some $k = O(n^{3/2})$, or for $k = O(n \log n)$?

A note which may help see the question as non-trivial: the fact that, for $k=o(n^2)$, the set $A^k$ need not act transitively on pairs of distinct elements of $\{1,2,\dotsc,n\}$ (see How many steps are required for double transitivity?) shows that we cannot always choose $w$ of the form $e_i\wedge e_j$.