Peroxide is not a good antiseptic. It is good for getting debris out, rinsing teeth, and for sterilizing surfaces (when in much higher concentrations than standard store bought) but it is horrible for basic wounds and especially wounds on something with thin skin and tissue. In study it did the most damage to healthy tissue of anything tested, slowed wound healing greatly, increased scarring, and I personally have seen the excess scar tissue it can create requiring it to be cut back down and the wound reopened to heal properly.
The best thing in studies and safest is chlorhexadine or betadine. I generally prefer betadine. If using repeatedly without bandaging or applying something to keep the tissue moist you may want to dilute the betadine to a weak tea color before rinsing the wound to help prevent over drying the tissue and again slowing healing. Otherwise it can be used to rinse a wound full strength before or between other treatments. With wounds that can be bandaged betadine can made in to sort of a poultice by soaking sugar in it to make sugardine and applying. Good for punctures and to help draw out abcesses. Another benefit is neither betadine nor chlorhexadine sting.