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 $$S=\{v \times w: v, w \in S\}?$$
1. **Claim:** All vectors in \( S \) have magnitude at most 1.

   *Proof:* Consider a vector \( v \) with maximum length in \( S \). Then, there exist vectors \( v_1, v_2 \in S \) such that \( v = v_1 \times v_2 \). Notice that \( v_1 \perp v \), so by maximality, we have:
   \[
   |v| \geq |v_1 \times v| = |v_1||v|
   \]
   This implies:
   \[
   |v_1| \leq 1
   \]
   Similarly, since \( v_2 \perp v \), we have:
   \[
   |v_2| \leq 1
   \]
   Therefore:
   \[
   |v| \leq |v_1||v_2| \leq 1
   \]
   \(\blacksquare\)

2. **Claim:** All vectors in \( S \) are either \(\mathbf{0}\) or unit vectors.

   *Proof:* Consider a nonzero vector \( v \) with minimal length in \( S \). Assume \( |v| < 1 \). Then, for any \( w \in S \), either:
   - \( v \times w = \mathbf{0} \), or
   - \( w \) is a scalar multiple of \( v \).

   If \( v \times w \neq \mathbf{0} \), then:
   \[
   |v \times w| \leq |v||w| \leq |v|
   \]
   By minimality of \( |v| \), this implies \( |w| = 1 \) and \( v \perp w \).

   Now, applying this fact to \( v \times w \), we find that for any \( w \in S \), either \( v \times w \) must be a scalar multiple of \( v \), or \( |v \times w| = 1 \). The latter is impossible by size, so \( v \times w \) must be a scalar multiple of \( v \), which is only possible when \( w \) is a scalar multiple of \( v \).

   Thus, every vector in \( S \) must be a scalar multiple of \( v \), but there will be no two vectors that cross to \( v \). \(\blacksquare\)

3. Hence, all vectors in \( S \) must be unit vectors. Consequently, they must be pairwise orthogonal. Now, it becomes clear that \( S \) must be one of the following forms:
   - \( S = \{\mathbf{0}\} \) (which has \( n = 1 \))
   - \( S = \{\mathbf{0}, \pm\hat{\mathbf{i}}, \pm\hat{\mathbf{j}}, \pm\hat{\mathbf{k}}\} \) (which has \( n = 7 \))

The final answer is \( \boxed{ n = 1, 7 } \)