The equality $L_{SO}=HOD$ can not be proved just in $ZF$. This is proved in the paper ``The consistency of the theory $ZF+L^1\neq HOD$'' by Szczepaniak.
Here $L^1$ refers to what you named $L_{SO}$. The idea of the proof is as follows:
$(1)$ If two models of $ZF$ have the same sets of ordinals, then they have the same classes $L^1,$
$(2)$ There are models $N_1 \subset N_2$ with the same sets of ordinals, such that there is a real $a\in N_1$ such that $a\notin HOD^{N_1}$ but $a\in HOD^{N_2}.$
Now the result follows from $(1)$ and $(2)$