When $n$ is even, then in some column there are at least two odd numbers. Then, the sum of pairwise product is odd so is not divisible by $n$. All $n$ odd work. To see it, write in $a$-th column respectively numbers $a, a, a \cdot \frac{n-1}{2}$ taken modulo $n$. Numbers in first and second row form a permutation, in third row too, because $\frac{n-1}{2}$ and $n$ are coprime