At f/2.4, the answer is NO, at least not for APS-C. Well, maybe it would be *possible*, but the optics wouldn't be real good. Wide lenses can be 1) small and fairly slow, or 2) big and fairly fast, or 3) have crappy optics. Compare the big DA14/2.8, the small DA15/4, and the tiny Fisheye Takumar 18/11.
For a camera with a smaller format than APS-C, it's no problem. Short fast lenses are common and fairly cheap, for small enough formats (like video). I have a lightweight 15-25mm f/1.2 zoom that cost me all of ONE BUCK! Only problem is, it's for a Super-8 camera. Doesn't work on my K20D, nope.
How-to build a fast wide lens: 1) take a FA50/1.4; 2) mount a 0.25x wide screw-on adapter on its front, giving a 13/1.4 optic; 3) ignore the distortions and aberrations and bulk and snide sneers. Better yet, just save up for the DA14 or DA15 I mentioned above. Or if you don't mind fishiness, get a Zenitar 16/2.8.