Existence of orthogonal matrices with zero diagonal and non-zero off-diagonal values

Question

Does there exist an orthogonal matrix whose diagonal values are all zero but whose off-diagonal values are all non-zero for any $\Bbb R^n$?

Furthermore, does this conclusion change if we are talking about unitary matrices and $\Bbb C^n$?

Answer 1

Hint Recall that the columns of an orthogonal matrix are pairwise orthogonal. So, for any $3 \times 3$ orthogonal matrix $A$ with zero diagonal entries, the dot product of, say, the first two columns is $$0 = (0, A_{21}, A_{31}) \cdot (A_{12}, 0, A_{32}) .$$