What you need to look at is the proof for isomorphism between the two groups:
Circle group - Wikipedia, the free encyclopedia
In a set of lecture notes on quantum mechanics it was stated that 'every representation of the rotation group is equivalent to a unitary one'.
I don't understand why this is the case, and I'd really like to understand how this works. Can anyone provide a proof of this or refer me to a source on this? I haven't been able to find a good source at all so far.