Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S\subset\mathbb{R}^3$ with exactly $n$ elements such that
$$S=\{v\times w:v,w\in S\}?$$ Please provide the sum of all possible values of $n$. 

Let's think step by step and output the final answer within \boxed{}.
user