Garlic burns occur due the noxious agents present in garlic that causes burns. These burns need to heal by themselves and cannot be healed soon. You can just aid in its early resolution by using virgin coconut oil and using oil enriched with vitamin E. You can also continue using cold milk and aloe Vera cream.
If you are worried about the pigmentation of the skin, I suggest you use whitening agents once the lesion is completely healed. There is no need for any antibiotics, until or unless you have a superimposed infection of that area. You can also apply local zinc oxide cream for better resolution of the lesion.
If the burn mark are very obvious and causing you excessive embarrassment, you can opt for laser removal of the spot which is very effective.