EDIT: broke this into two parts - for these were two different questions.
For numerically obtaining the stabilities of a matricial equation, i need to generate an ensemble of matrices that are members of SU(n), and strategies for uniform random generation of those are outlined here. But I also need these unitary matrices to be clustered close to the SU(n) identity operation, at some fixed "distance". Thus my question distills to -
What might be a good metric for SU(n) matrices? EDIT: one option has been provided by @Jim_Belk downstairs (:D) - certainly feel free to add more!
How does one generate SU(n) matrices close to the Identity?
EDIT: are there any alternatives other than computing the generators first and then exponentiating them?
If you find the questions framed improperly or dubious, do tell!