Any wound of significance should be dressed. Multiple studies have shown that a wound that dries out heals slower than one that is covered especially if an antibiotic salve is applied. The physiology is that once a wound dries there are two processes that interfere with closure. The first is that the scab must be pushed out of the way in order for new tissue to grow in its place, and a dry scab is harder to extrude. This is why we have the itch reflex. Animals need to lick or scratch any dry scab out of the wound to allow for healing. The second process is that newly developing tissue has not been covered with keratin, the protective layer that prevents drying, and can therefore dry out this creates a callous of dead skin instead of pink healing skin.
Synopsis; The old addage of letting a wound dry out and get air is bull.