Proof that a specific determinant involving the floor-function is non-zero

Question

Let $p>5$ be a prime. For any positive integers $1\leq i,j\leq (p+1)/2$ let $ a_{i,j}=\lfloor ij/p \rfloor $. It is a fact that the determinant $|a_{i,j}|$ where $3\leq i,j\leq (p-1)/2$ is non-zero. The known proof is very indirect and sophisticated. Can you give an elementary proof?