Yes, but TripleR only fits up to 2 variables. So if you plan to do this for all Big-5 factors, you would have to run bivariate models for each pair to get results.
Nestler (in press) proposed multivariate SRM for an arbitrary number of variables using (RE)ML estimation
He provides R syntax for fitting it here:
But I don't know whether correlations with self-ratings are already included in that procedure.
You shouldn't need to export anything, if self-ratings are included on the diagonal of the matrix(es) submitted to RR(). Look at the ?selfCor help page, or read the instructional document linked on the ?TripleR help page.
Terrence D. Jorgensen
Postdoctoral Researcher, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam