Burns often do well with a vaseline gauze type dressing, under melolin. Much less adhesive than melolin on it's own, and keeps the wounded area nice & moist, while clean – this is certainly the bog standard dressing that my local burns & plastics unit apply, unless they've changed very recently.
Other time to seek help with a burn is if it is circumferential (?sp) as the risk of swelling may restrict circulation. Doesn't sound like the case in your injury, but be aware of any "flu" type symtpoms coming on, I've seen some cases of toxic shock from very innocent appearing injuries that have been neglected.