Unless you stabbed yourself, that probably isn't that deep. Looks to me like you got to the layer of subcutaneous fat, but no farther. If you clear the blood, you'll probably see a white layer underneath.
Yeah, no. Peroxide is generally not a good astispetic for cuts because it also destroys healthy tissue,which leads to increased scarring and healing time. Alcohol would be a better choice. Neosporin isn't particularly good for deeper wounds either. If you get a scrape or burn or something, it's okay, but I wouldn't use it for anything that goes beneath the skin. My advice about closing it up still stands. Otherwise you'll end up with a raised, reddish lump that would probably take about a year (or longer depending on how you heal) before becoming white scar tissue. If you pull it shut with butterfly closures or even tape/whatever, this will be a lot faster, and the scarring will be much less noticeable.
I unfortunately have a lot of experience with cuts about this deep, so I'm speaking mostly from experience.