I use hot water to rinse the soap off the razor as I’m shaving. There might be a little bit of residue that builds up, I try to hit it with some hot water. If it’s my straight razor I sometimes carefully wipe the blade with a towel before stropping (this helps dry it and prevent rust, as well as remove any remaining soap scum).

On DE safety razors I take a q-tip and sterilize the blade with 70% isopropyl alcohol which happens to also clean the soap scum off. I sterilize before shaving to help prevent infections to the follicles that can be caused by bacteria that start to live on your razor. Another recommendation I’ve heard is just to store your razor in 70% alcohol, but I prefer my way.