Itai,
thank you very much! In exactly one line what you wrote makes the job. Actually I did not understand very well what you wrote, but I am going to study it!

That's right.  For a normal matrix, the Schur decomposition and spectral decomposition coincide.  Our Schur decomposition is limited to approximate numeric matrices, so depending on whether support is needed for exact and/or symbolic inputs, it may be a good choice.  realDiagnolization, above, uses the fact that we have a very specific matrix type to provide an answer that works for symbolic and exact numeric inputs as well.

There is no builtin function which does this, but it isn't hard to construct one yourself. The basic process is quite similar to the normal diagonalization of matrices; the key point is that the real and imaginary parts of the eigenvectors are part of the same two-dimensional subspace. Thus, rather than constructing the orthogonal matrix out of eigenvectors, you use the real and imaginary parts from each complex-conjugate pair. Since the function Eigenvectors keeps the conjugate pairs together, you can do this:Here [[1;;-1;;2]] selects only the odd eigenvectors, Map (/@) is used to create the real and imaginary parts, Flatten is used to ensure we get n vectors (instead of n/2 pairs of vectors), and Normalize makes them all unit. You can then construct the diagnoal form as For example, consider this matrix:We have (I'm using N and Chop to create a numerical approximate that fits on a screen):And
This is in fact the matrix returned by Jose's CanonicalAntisymmetricMatrix function

I don't think it's any of them...