Denote by $\DeclareMathOperator{\uso}{\underline{so}}$ $\uso(n)$ the space of symmetric $n\times n$ matrices. $\DeclareMathOperator{\rank}{rank}$ Note that for $A\in \uso(n)$
$$ \rank A=n-\dim\ker A, $$
$$\dim \ker A \equiv n\bmod 2. $$
Denote by $ \uso(n)_k $ the space consisting of matrices $X\in\uso(n)$ such that $\dim \ker X= k$, where $k\equiv n\bmod 2$.
You are interested in the space $\uso(n)_{n-4}$.
We have a diffeo $\uso(n)_k\to\uso(n-k)_0$ which associates to a matrix $X\in\uso(n)_k$ its restriction to the orthogonal complement of the kernel. We deduce
$$\dim \uso(n)_k=\dim \uso(n-k)_0=\dim \uso(n-k)=\binom{n-k}{2}. $$
Thus, in your case
$$\dim \uso(n)_{n-4}=\binom{4}{2}= 6. $$