  We derive explicitly the structural properties of the $p$-adic special
orthogonal groups in dimension three, for all primes $p$, and, along the way,
the two-dimensional case. In particular, starting from the unique definite
quadratic form in three dimensions (up to linear equivalence and rescaling), we
show that every element of $SO(3)_p$ is a rotation around an axis. An important
part of the analyis is the classification of all definite forms in two
dimensions, yielding a description of the rotation subgroups around any fixed
axis, which all turn out to be abelian and parametrised naturally by the
projective line.
  Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$
admits a representation in terms of Cardano angles of rotations around the
reference axes, in close analogy to the real orthogonal case. However, this
works only for certain orderings of the product of rotations around the
coordinate axes, depending on the prime; furthermore, there is no general Euler
angle decomposition. For $p=2$, no Euler or Cardano decomposition exists.
