Suppose we have a (say, ordered) set $$(X_1, \ldots X_k)$$ of pairwise orthogonal vectors, say with span $S$. This determines a set $$(Y_{ij})_{1 \leq i < j \leq n}, \qquad Y_{ij} := X_i \wedge X_j,$$ which by construction is a basis of the space $\Lambda^2 S$ of bivectors.
Is there a sense in which the bivectors $Y_{ij}$ are themselves (pairwise) orthogonal?