It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a corresponding statement for non-orientable closed surfaces of negative Euler characteristic. By the Gauss-Bonnet theorem and Ricci flow theory, for example, I can see that such surfaces admit many hyperbolic metrics, but I am not sure about the structure of the space of such metrics.
Also, on the Teichmüller space for a genus $g \geq 2$ closed orientable surface, there is a concept of distance (see, for example, Definition 6.4.1 of Buser's book "Geometry and spectra of compact Riemann surfaces"). Is there a similar concept in the non-orientable case? Thanks in advance!