Engine fan runs constantly

Hi, I have a 1996 Dodge Intrepid, 3.5 L 6 cyl, 163,000 miles. Yesterday, I suspected the engine fan was running at longer intervals than normal while the car was idling, so I had my husband check the coolant level. The coolant level was fine; however, this morning after driving approx. 12 miles to work I noticed steam coming out of the hood. When I got to work the coolant was pouring out from the cap. The cap looked like it was not screwed on properly. The engine temp. Gage did not indicate the engine was overheating. After the engine cooled for approx. 1 hour, I poured a gallon of antifreeze 50/50 mix and some water in the coolant tank. I took it for a small test run and the engine seemed to be running at the normal temp, but the fan runs all the time now. I drove it approx. 2-3 miles at lunch and the fan stayed on from the time I started the engine to when I shut if off. Do you think this is from the coolant leaking problem I had or is there possibly another problem? I appreciate any advice.