Hello,
No, you will not get a W-2, you should have received a specific form from Social Security , a SSA-1099. It comes as one sheet of paper that you have to unfold to open and read. It looks like the annual "Your New Benefit Amount" notice but has the words "FORM SSA-1099" at the top. I am not sure if you will get one or two different SSA-1099s.
I hope this helps.