In upper half space, the thing that's analogous to the invariant 2-form (which measures hyperbolic area) is the hyperbolic volume form $1/z^3 dx dy dz$.
If you want to understand 2-forms invariant by a particular discrete group, they are the same thing as 2-forms on the quotient orbifold. In the case of $SL_2$ of the algebraic integers in a quadratic imaginary field such as this, the groups are usually called Bianchi groups: Bianchi worked out fundamental domains for them for the first set of examples, as well as a general technique to find fundamental domains, On Allen Hatcher's website, there's a handy list of pictures of the folded-up fundamental domains in cases where they can be readily drawn. In this case, the quotient orbifold has underlying space $S^3$ minus a point (for the cusp), with singular locus in the form of the 1-skeleton of a tetrahedron. Since the underlying space is contractible, any closed 2-form is a coboundary --- \emph{i.e.} there are plenty of 2-forms, closed or otherwise, but they're no obvious reason they should be interesting. In this particular case, the diagonal matrix with entries $i, -i$ rotates 180 degrees about the $z$-axis, so it takes the 2-form above to its negative --- so it's not invariant even before attempting to extend it.