Test your alternator with the car running by putting a multimeter on your battery. The reading should be somewhere between 14 and 15 volts DC.
If that's fine, then stop your engine and if your multimeter measures current, then test to see if your positive battery wire is drawing current with the cars off, doors shut, any under hood lights off, etc. Basically, make sure nothing is drawing current to see if you have a parasitic electrical leak, then diagnose from there by pulling fuses until you figure it out. Might be something quick and easy, but also might be a PITA.
Also, your dash may indicate if your alternator has failed through voltmeter or alternative warning light. Never keep a dead/low battery in the cold, or it will freeze.
It is a good practice to throw a charger on your vehicle battery once every 1 to 2 months for an overnight trickle charge, and especially in the winter months when it gets really cold as it has been. Most alternators are good at keeping a battery operational, but they are never sufficient to bring a battery up to a full charge. A less than full charge is hard on your battery because it makes it work harder in the winter when it is cold, and it has a higher risk of freezing. A less than charged battery also causes precipitation on the lead plates, which leads to battery failure. There are special trickle chargers that will ensure that a battery will be reliable over the long term.
The optima 400 are pretty awesome chargers if you have the cash. They can recover a dead battery as low as 1.5 volts.
https://www.optimabatteries.com/en-us/battery-charger