I have replaced the tube on the rear tire of my fixie and every time I go out for a ride, the tube bursts after 20 minutes. I was wondering if this could be because I skid on the worn spot on my back tire, or if im pinching the tube while trying to install it. Should I just buy a new back tire?

Buying new stuff w/o being sure of the cause of the failure is usually not such a good idea. A bit more bike forensics is advised. Burst implies that the tube is literally split open along a considerable length, and is commonly caused by pinching it under the bead.
It shouldn't be caused by a worn spot, unless:
- the spot is worn through, allowing the tube to bulge outside the profile of the tire.
- Or if what you're seeing isn't a burst, but a puncture. And if the tire is really thin at some point(s), it will be more sensitive to punctures caused by road debris on that spot.

Hmm is the puncture in the same spot each time? If so check for sharp bits on the rim, or something stuck in the tyre.
but it is probably getting pinched. If you can, install it without tyre levers, just using your thumbs. And always put a bit of air in the tube, enough that it has a shape. Once everything is in place, go around the circumference pinching the tyre and looking to see that the tube is not pinched between the tyre and rim.
hope that helps

Be careful to seat the tire bead fully without pinching the tube between the rim lip and the tire bead. Before inflating to pressure, work around the rim with your thumbs, pressing the bead inward and peeking along the inner rim lip. You're looking to make certain that you don't see any of the tube showing. Work around both sides evenly, looking carefully. Then, you're good to inflate.

Also, following a blowout, be careful to do your forensics. Mark WHERE the the tube puncture corresponds to the rim and tire BEFORE removing fully. What can you see from the tube? Is it a long tear, indicative of a bead-pinch or bulge blowout? Or a shorter or pinpoint, indicative of puncture? Are there a pair of small holes, like a snake fang? That's an under pressure pinch flat. Examine the tire at the point of puncture. Is there embedded debris on the inside? An exposed/thin tread area that is subject to debris puncture? Something else?

Why do you skid on your tires? It winds up accelerating their wear and costs money...

why do anything fun?

Burst or blowout is generally taken to mean an explosive flat - immediate and loud. If that is what's happening it ONLY occurs when the tube first escapes the confines of tire and rim - either due to poor installation technique or a fault/defect in the tire. If not as noted you need to look at where the flat occurs and what the puncture looks like in order to prevent more occurrences.