Does it exists any $n(n>2)$ dimensional orthogonal matrix with exactly one zero in each row(column)? How to construct a general one if it exists?

Question

I am wondering whether there exists a $n(n>2)$ dimensional matrix with exactly one 0 in each row(column), I will be appreciated if anyone can provide a solid example for general $n(n>2)$ or give a proof that it does not exist, thanks!

Sorry for misdescribing my question, I am asking whether that statement holds for general $n(n>2)$. Or if the statement holds true, is there any general methodology to construct this kind of matrix for any $n$?

Answer 1

How about $$\frac1{\sqrt3}\pmatrix{0&1&1&1\\1&0&1&-1\\1&-1&0&1\\1&1&-1&0}?$$