No reference site I found seems to list these (surprisingly), so you're just going to have to find the solid angle yourself: http://mathworld.wolfram.com/SolidAngle.html
You don't need integration, just vector calculus. I don't think it's hard, just obscure.
This guy derives the following vector formula for the solid angle of a (skew) tetrahedron:
http://answers.google.com/answers/threadview?id=160247
You can use that subresult, decomposing the square base of the skew pyramid into two (possibly axially symmetric) triangles:
Let U,V,W be three vectors emanating from the origin, with respective lengths u,v,w. The (signed) area corresponding to the solid angle formed at this corner is:
2 arctan( det(M)/(uvw + u V.W + v U.W + w U.V) )
where M is the matrix formed by stacking U,V,W as rows and "." denotes vector dot product.
I think you can formulate the sum of solid angles subtended by two base triangles quite easily.
They got that result from MathForum: Geometry Puzzles (http://mathforum.org/kb/forum.jspa?forumID=129)
That should be everything you need.
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Other than that, I did search for "solid angle skew pyramid" and found some comments on the mailing-list of:
AntiPrism (open-source software)
http://tech.dir.groups.yahoo.com/group/antiprism/message/3416
http://www.antiprism.com/
You might try searching or asking on their mailing-list.