Relationship between the entries of a rotation matrix, Mathematics

1. A 3d rotation matrix has 9 (3 by 3) entries, and a 2d rotation matrix has 4 (2 by 2) entries. How many actual degrees of freedom are there in a 3d or 2d rotation? In other words, what is the minimum number of parameters necessary to uniquely specify a 3d rotation and a 2d rotation (answer with 2 numbers)? What does this imply about the relationship between the entries of a rotation matrix?

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