[Math] The rank of skew-symmetric matrix is even

Question

I know that the rank of a skew-symmetric matrix is even. I just need to find a published proof for it. Could anyone direct me to a source that could help me?

Answer 1

Notice that over a field of characteristic 2, the matrix $$\begin{bmatrix} 1&0&0\\0&1&0\\0&0&1 \end{bmatrix}$$ is skew-symmetric, and has rank $3$. So I think you are really looking for a proof that an alternating matrix (skew-symmetric with zeros on the diagonal) has even rank.

The proof for this usually comes up in looking at the theory of bilinear forms, there are a few references mentioned in the comments that contain proofs, but another option is in "Classical Groups and Geometric Algebra" by Larry Grove.