# How many steps on $S_n$ are required to span $V\wedge V$, $V = K^n$?

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 $$e_i\wedge e_j$$.