Suppose I want to generate a $n \times n$ orthogonal matrix $H$ (that is, $H^T H=I$) but with the property that $1-e < (tr H)/n < 1+e$ for some pre-specified tolerance $e$. How can I do this? Ideally I would want them to have the Haar distribution (conditional on the trace restriction) but I would settle for any method that produces a "reasonable" distribution.
Two approaches I had in mind:
1) use of Givens rotations/Euler angles close to 0
2) (edited) using skew-symmetric matrices $L$ close to 0 and evaluating $\exp L$
NOTE: Possible applications include Bayesian estimation of covariance matrices.