Is there a way defingin the Y axis of a sketch?

It should already be defined for you (longer red arrow).

Can you attach more of a description of your specific need?

Are you looking for a permanent Y-axis Feature to reference all sketches?

In this case, most users create this from intersection of origin planes and then save it as template (along with x and z axes.)

15 years ago, this was a deficiency.  Now it's a long-standing tradition.

Peter,  I believe you're looking for the modify sketch tools command.  One prerequisite to this is you have to have at least some entity in your sketch to make the 'modify' command not be grayed out (weird, not sure why).  Now you can right click on the nodes of the black csys that pops up to flip (mirror) to other orientations or you can plug in angular dimension in the menu that popped up to change the sketch orientation.  I hope this helps.