I hit my elbow hard and now when I lean on it, I get a sharp shooting pain. It seems to be getting worse. Is there something I can do at home to help?

Yes. If you injured the ulnar nerve which runs thru the elbow in the cubital tunnel then you need to get an elbow pad to avoid putting pressure on the nerve. If there is any concern for a fracture then please see your primary care doctor to get an x-ray. Good luck.
May heal slowly. Some people use elbow pads but most just learn to avoid leaning on the sore elbow until it heals. Elbow injuries often involve bone bruising, and that can take months to heal. If you might have hit it hard enough to break a bone, an x-ray is a good idea.