Skew matrices generate everything? (This is a reformulation of my previous question with a little less notation and theory.)

Let $K$ be a field of characteristic zero. Consider the matrix algebra $A = M_n(K)$, and let
  $$
S = \{X\in A\mid X^T = -X\}
$$
  be the set of skew matrices in $A$. Does $S$ generates $A$ as an algebra with unit? 

It is obviously false if $n = 2$, but does it become true for $n\ge 3$ or for even larger $n$?

Let $e_1,\ldots,e_n$ be the canonical basis of $K^{n}$ (they are column vectors). Notice that $\{e_{i}e_j^t-e_je_i^t,\ 1\leq i,j\leq n\}$ generates the subspace of skew symmetric matrices. 
Now $(e_{i}e_j^t-e_je_i^t)(e_{j}e_i^t-e_ie_j^t)=e_{i}e_i^t+e_je_j^t$ and the set $\{e_{i}e_i^t+e_je_j^t,\ 1\leq i,j\leq n\}$ generates the subspace of diagonal  matrices when $n\geq 3$. 
Finally $(e_{i}e_i^t-e_je_j^t)(e_{i}e_j^t-e_je_i^t)=e_{i}e_j^t+e_je_i^t$. Now the set $\{e_{i}e_j^t+e_je_i^t,\ 1\leq i,j\leq n\}$ generates the subspace of symmetric matrices. 
So the algebra generated by the skew-symmetric matrices is $M_n(K)$.
