One of these days I need to do a video on wound care in the woods. Once you've broken the skin, there is no such thing as a sterile wound, even in surgery. There are bacteria and other organisms on your skin even after we've "prepped" it with betadine, alcohol scrub, chlorhexidine etc. But even though we never get a completely sterile wound, the wound infection rate for a "clean" wound in surgery is only 1-4%. It can be even less depending on the type of surgery and the body part operated on. I have a zero percent infection rate with surgery on the face after 5 years of doing so.
In the woods, you're dealing with a contaminated wound. The thing to do is stop the bleeding with direct pressure and then wash the wound with water that you would drink. If you can drink the water, you can wash a contaminated wound out with it. Get all of the dirt and other debris out of the wound. If it starts bleeding again, hold pressure. Seek medical attention as soon as possible.
Bandage requirements are much too complex for me to get into in a short post, but suffice it to say that the bandage will vary based on the wound itself and the environment you're in. Using no bandage at all is acceptable depending on the wound itself and the environment you're working in.
If you want to put something on the cut, put vaseline on it, unless you're allergic. Do this ONLY AFTER the wound has been cleaned.
There is some good information in this thread. There is also some very very bad information in this thread. Reader beware.
This is the example of good information