Find rank of very big matrix.

Question

Find rank of $$\begin{bmatrix}
0&1&1\ ..&1&1&1\\
-1&0&1\ …&1&1&1\\
-1&-1&0\ …&1&1&1\\
..&..&..&..&..&..\\
-1&-1&-1\ ..\ &-1&-1&0
\end{bmatrix}$$

After row elimination I get this \begin{bmatrix}
1&0&-1\ ..&-1&-1&-1\\
0&0&0\ …&1&1&1\\
0&-1&-1\ …&0&0&0\\
..&..&..&..&..&..\\
-1&-1&-1\ ..\ &-1&-1&0
\end{bmatrix}

but don't know how to continue.

I know how to find rank by minor but I don't think we need it here because we can't check every matrix whose determinant is not $0$. Therefore we need to find rank by row elimination.

Answer 1

Hint: Denote the skew-symmetric matrix by $A_n$. For $n$ even the determinant is nonzero, namely $\det(A_n)=1$, so the rank is equal to $n$.

Reference:

Determinant of a special skew-symmetric matrix

For $n$ odd the rank is $n-1$ and of course $\det(A_n)=0$.