It is hard to predict exactly how long the skin will take to return completely to normal. Based on your description, it does sound like the injury
so far is healing as expected. With scrapes like these, the scab tends to stay in place for a few weeks, after which it will fall off to reveal the new skin, which tends to be initially bright pink in color but may change to a brownish color as you have noticed as well. However, talking to your doctor
will help confirm what is going on.
First, the most important thing to speed healing is to avoid picking at the scab. Premature removal of the scab will result in delayed healing and potentially scarring as well. Applying Neosporin, as you are doing, may also help both by preventing infection and also keeping the new skin moist, which speeds healing. A band-aid may also help with keeping the area moist and protected.
The brownish discoloration that results at the end of the healing process is called post inflammatory hyper pigmentation. This, unfortunately, can last quite a while, up to several months, and it is difficult to predict exactly how long.
I would also show this injury to your primary care doctor
, who can provide you with additional advice and make sure that healing is progressing as expected.