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

$$\dim \uso(n)_{n-4}=\binom{4}{2}= 6. $$