Frankly speaking 대한민국 already states, that this is South Korea, not North.
North Korea has a bit longer name 조선민주주의인민공화국,
but there are not so many people with Internet access there.
So, of course you can change English name to South Korea.
Yuri.
I recommand ROK(Republic of Korea) than south or north.
Of course so many people use south korea than ROK… but it is not an official one.
(As DPRK is not north korea)