First problem: Blood is an organic substance, and like all organic substances that aren't cooked, refrigerated or otherwise preserved, it will rot. You'll need to freeze or at least refrigerate it.
Second problem: It is a biohazard. Any diseases the donor had (known or unknown) will potentially be in that blood at infectious levels. Any bacteria introduced during or after collection will also grow very nicely in it unless it's refrigerated or frozen. You really should put a biohazard warning label on your art so people know it is potentially infectious. Keep in mind that some viruses can remain infectious in dried blood for weeks (eg, hepatitis B).
Third problem: Blood clots and congeals. After a week you won't have something you can paint with, and it will be black, not red.
You could try adding a chemical such as chlorine bleach or formaldehyde as a preservative, but I'm not sure what other effects that would have.