PMO 2018 Problem 3

Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [] the $n^2$ entries are integers from $1$ to $n$ ; [] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$ , such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.