Unfortunately you can not contact Coach to verify the serial number on your bag. Coach recommends buying their merchandise from either a Coach store or an authorized retailer.
Having a serial number does not mean the bag is real, as counterfeiters put serial numbers on their bags as well. In general, the number should start with a letter from A to M. If the bag does not have a serial number, it might still be real, but it could just be an older model.
Fake Coach bags will often have irregular stitching, breaks or misalignment in the Coach pattern, or inferior materials.