Tensor product of half-spin representations

Question

Let us consider half-spin representations $S^+$ and $S^-$ of the orthogonal Lie algebra $\mathfrak{so}(2n, \mathbb{C})$. It is known that $\bigwedge\limits^{n-1}V$ arises as subrepresentation in $S^+\otimes S^- $. Is there canonical description in terms of spinors of this subrepresentation? (May be as kernel of some pairing?)