Yes. Menstrual cycles are so different is every woman, but you also need to make sure you are not becoming anemic or developing a nutritional or mineral deficiency when you menstruate. Vitamin or folate (folic acid) or iron deficiency can all cause gingival (gum) pain. See your ob/gyn and possibly consider further work up to ensure no deficiency exists.
Hormones. Hormones affect gums, that is the most likely gums. The healthier your gums are the less likely that is to happen. Do you have regular professional care. If you do not, please seek it out.
Normal but... Some women are very susceptible to oral complications from hormonal fluctuations. Ideally, you should have your teeth professionally cleaned and double your efforts at home in terms of brushing and flossing. Bleeding means you need to do more fast.
Sore Gums. Sore gums mean irritated or inflamed gums. This may happen briefly during the mid-cycle hormonal surge, but is uncommonly the cause of sore gums. Go see your dentist, if this symptom doesn't clear or recurs to make sure of the cause.
Very common. Menstruation gingivitis usually occurs a day or two before the start of the period. Keep a good oral hygiene and visit your dentist on regular basis.