Let $V$ be a $6$-dimensional subspace of the space $\mathbb{R}^{3\times 3}$ of real three-times-three matrices. Is the intersection of $V$ with $O(3) \subset \mathbb{R}^{3 \times 3}$ always non-empty?

If not, how can we characterize those subspaces $V$ such that $V \cap O(3) = \emptyset$?

\Edit: Anton made the trivial comment below that one can take matrices with first (or some other) column equal to zero. However, how can we characterize such subspaces?