Your first and third cases don't take much doing. As you can see directly by the power series expansion, $\exp(0)=I+0+\frac{0^2}{2}+...=I$. You can see that exponentials of diagonal matrices are simply the exponentials of diagonal entries, so that we can find a logarithm of any exponential matrix, for instance your $S$, by taking logs of each diagonal entry. You'll notice in particular that $S$ only has a logarithm over the complexes. This will generally be true when we have negative eigenvalues in a real matrix.
As to the anti-identity matrix, things get a bit more complicated. These are all symmetric, and so they're unitarily diagonalizable over the complexes. Then you may write $T'=V^{-1}TV$ for $V$ unitary, and $\log(T)=V^{-1}\log(T')V$, where we've already seen how to get a logarithm of a diagonal matrix. Yet I'm not sure you'll find an expression for this in arbitrary dimension, if that's what you're after.