Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.

For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three matrices, consider its intersection with $O(3)$.

• Is it generically true that $V \cap O(3)$ is non-empty (i.e. the set of such $V$ is an open subset of $\mathrm{Gr}(3, 6)$)? Can one say how many points the intersection generically includes?
• Is it possible to characterize the $6$-dimensional subspaces $V$ such that $V \cap O(3) = \emptyset$? (For example, if $V$ consists only of singular matrices, then $V \cap O(3) = \emptyset$; conversely, does every $V$ with $V \cap O(3) = \emptyset$ consist only of singular matrices?)

Old thread 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?