Eczema?
 

I have Eczema on my wrists and hands, and usually it heals up and goes away after a while. But this time, it won't. Is there a way to make is heal faster? Currently the backs of my wrist is bleeding and looks infected from this, and I really don't know how to treat it myself.

You need a good moisturizer like aqueous cream. spread it on thickly. You might even want to put a pretty thick layer on an bandage the area to stop it coming off.
It might also be worth speaking to your gp as they can prescribe a better moisturizer as well as checking that it is not infected.

I have had eczema most of my life. It gets worse in hot conditions. Especially during the night. So during the night maybe put a stretchy bandage on it or keep your room cool (by windows or a fan). That should help. Also, you should go to your doctor and they will give you a moisturizer such as Aveeno or a steroid cream such as Lociod Lipocream or Elecon.
If not, just apply plenty of moisturizer, especially after baths, showers or washing your hands (anything that involves getting the eczema wet.)
Hope that helps.

If like you say it is infected then there is nothing you can do but see a doctor and with skin infections it can take a long time for them to heal so it may be they advice you to use antibiotic/ steroid creams on it as well as taking antibiotic tablets which could be for a few weeks.

Really if it starts to flare up you should use aquesous creams under layers of bandaging changing it twice daily but if it is worsening often and you don't know how to treat it this should be discussed with your GP/ dermatologist.