I did the quest to get the ring of Hircine but I thought I was meant to kill Sinding. Now I have completed the quest and only have his skin. Is there any way that I can re-do the quest or go on another quest to get the ring again?
For your question, If you got the skin first, you can't get the ring afterwards.
However, there are some instances where you can exploit a glitch to get more than what was intended.
If you (somehow) are able to load an old save before the start of the mission, you can get the ring first, then at the end of the mission, you can get the skin (from my previous answer):
Sometimes, the game glitches out and upon completion of the quest (when trying for the ring) Sinding will reappear in the cell you first found him, in werewolf form. Killing him will relent the Savior's Hide. (this is tricky through the bars of the cell, but with timing and a save beforehand, you should prevail)