You're set up and focus sounds good.
Once I get set, i try to relax my grip on the bar. In the right position (low-bar for me) I don't need to "hold" the bar at all. My hands are more like stops/catches, but not gripping the bar. If you torque the bar, you may be straining the elbow. If not that, try a different hand position (probably wider) to see if it alleviates the problem.
Some bicep, tricep and shoulder warm up and light stretching are also a good idea before squats.