Using penicillin on everything is how antibiotic resistant infections occur.
If the dog can lick it, leave it alone.
I had one get a 4" deep! X 2" long puncture/slice in her ham right by her twat just 3weeks ago and followed this ancient advice and now it's totally healed up already and she's been back at full speed since a week after the cut.
Different strokes for different folks.
X2 If they cant lick it, we leave it alone. Had one cut in the flank just as you described and it was healed up in two weeks. The only thing I did was wash the wound with a water hose every few days.