Make sure you're wearing good gloves. Gloves aren't just for regular comfort and to avoid "hamburger hands"...

There are several important little tuberosities on your carpals (wrist bones), which is where most of your weight is carried if you're riding with a flat bar. Irritation of your carpals can cause a lot of radiating pain-- which can refer up the medial aspect of your forearm...

Otherwise, it's possible you might be a bit too far from your handle bars if you're keeping your elbow locked. You may just need to strengthen some of your forearm flexors by paying close attention to that flexion.