When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by removing the $i$-th line and $i$-th column of $M$, multiplied by $(-1)^i$.

Example with $n=3$: $$\begin{bmatrix} 0 & a&b \\ -a&0&c \\ -b&-c&0\end{bmatrix} \begin{bmatrix} c\\-b\\a\end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}$$

Example with $n=5$: $$\begin{bmatrix} 0 & a & b & c & d \\ -a & 0 & e & f & g \\ -b & -e & 0 & h & i \\ -c & -f & -h & 0 & j \\ -d & -g & -i & -j & 0 \\ \end{bmatrix}\begin{bmatrix} -e j+f i-g h \\ b j-c i+d h \\ -a j+c g-d f \\a i-b g+d e \\ -a h+b f-c e\end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$$

I suppose this is a known result: would you know a reference which mentions it?