There are so many options in that price range. The question you should really be asking yourself right now is what type/size of knife are you wanting. Do you want carbon or stainless steel?
For "splitting wood", I'm assuming you mean by batoning, because most knives aren't very good at splitting (to short and to narrow). So if you're wanting to do stuff like that, you should probably be going with something full tang, and if it were me, it would be in a carbon steel. I'd also be looking at something with a thickness of somewhere in the range of 1/8in and 3/16in. And for a fixed blade, I like them at least 4inches. Beyond that though, things change depending on what length the blade is, because their purpose changes.
I use my Becker BK9 like a hatchet (chops better than my lame hatchet), and baton it to split stuff that approaches forarm/calf sized wood. I cut that wood with a folding saw though, because chopping just isn't worth the effort imo. I use my Mora for feather sticks, and finer tasks (its just barely 4in of blade). There are a bunch of blades in between (and further up or down the size "scale"). If you want a SINGLE blade to do everything, then you should be looking at the 5-7in range (blade length). I'd start with the ESEE 6 or Becker BK12 personally (both are 6in blades). But it really comes down to what you're wanting out of it.
So any ideas on your preferences in materials, and size?