I wasn't able to export a skydome from UT3 for you to look at and analyse, their packages are "cooked" and won't let you export them. So here's the next best thing (I hope), some screenshots;
The skydome from the UT3 level "Islander" is basically a dish shape. The main part, the top, is like a convex circle, whereas the sides are like a cylinder shape. This is a more effective shape than a semi-sphere or a box, since it compensates for the player looking in the horizon.
Here is the texture file for that skydome. The top part is flat, whereas the sides are warped, much like the shape you got in photoshop Craig. You should be able to save this image (it's only 512, unfortunately, I couldn't fit the 1024 on screen to take a screenshot.) and try it out in Max on a simple mesh.