How do I make and setup a correct image texture for reflection mapping in BGE?

I want to take advantage of realtime reflection mapping in the game engine. Specifically, I’d like to create a skymap image that some object materials will reflect, so as to fake a glossy shader (just like Blender’s matcap feature in the 3D view) However, I’m not sure how to correctly render such a skymap, and set it up so that the material reflects it correctly without distortion.

As you can see in this initial setup, a flat image is heavily distorted along the edges. I tried rendering a 180 fisheye panorama in cycles but this did not produce satisfactory results either.
What can I do to get the best, most correct results with reflection mapping?