Can any rotation be a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?

The rotations on $\mathbb R^3$ have the structure of  a group and are not a vector space, so rotations cannot be vectors. This is because the combination of two rotations depends from the order (it is not commutative) so it cannot be represented as an addition operation of vectors.
As you noted we need three numbers $(v_1,v_2,v_3)$ for a vector that identifies the axis of rotation with the condition that te vector is normalized: $v_1^2+v_2^2+v_3^2=1$ and a fourth number for the angle of rotation around this axis. So we need four numbers with a condition about tree of them.
These four numbers can form the objects of different mathematical structures that are possible representations of the group of rotations. Possible representations are the $3\times3$ orthogonal matrices with unit determinant or the  unit quaternions, that are a generalization of the complex numbers with four components.
