  Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code
$\mathcal{C}$ $(k = 3, 4)$ into a self-orthogonal code of the shortest length
which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$.
We extend this result by proposing a new method related to a special matrix,
called the self-orthogonality matrix $SO_k$, obtained by shortening a
Reed-Muller code $\mathcal R(2,k)$. Using this approach, we can extend binary
linear codes to many optimal self-orthogonal codes of dimensions $5$ and $6$.
Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by
showing that if $31 \le n \le 256$ and $n\equiv 14,22,29 \pmod{31}$, then there
exist optimal $[n,5]$ codes which are self-orthogonal. We also construct
optimal self-orthogonal $[n,6]$ codes when $41 \le n \le 256$ satisfies $n \ne
46, 54, 61$ and $n \not\equiv 7, 14, 22, 29, 38, 45, 53, 60 \pmod{63}$.
