Jim, I believe you can't obtain a signed area of a 2D triangle in 3D space without adding extra structure. From what vantage point are you looking at the triangle? What happens if you flip the triangle over? Would the sign change at some point?

What you are talking about is an orientation. An orientation only applies to an object that spans the entire space. You need to define an extra vector field that gives an orientation at each point of the 3D space. For example if the triangles were only allowed in a given plane then you could define an "outward" normal for the plane and use it to give an orientation (or sign) to all triangles in the plane. If you had a sphere and the triangles were in the tangent planes of the sphere you could use the outward normal vector from the center of the sphere to establish an orientation. If you had a convex polyhedron you could use outward normal vectors from some point within the polyhedron.

And, of course, not all surfaces can be oriented.