  We provide a general expression of the Haar measure $-$ that is, the
essentially unique translation-invariant measure $-$ on a $p$-adic Lie group.
We then argue that this measure can be regarded as the measure naturally
induced by the invariant volume form on the group, as it happens for a standard
Lie group over the reals. As an important application, we next consider the
problem of determining the Haar measure on the $p$-adic special orthogonal
groups in dimension two, three and four (for every prime number $p$). In
particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a
direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$
and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on
these two groups can conveniently be lifted to Haar integrals on certain
$p$-adic Lie groups from which the special orthogonal groups are obtained as
quotients. This construction involves a suitable quaternion algebra over the
field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the
real rotation groups. Our results should pave the way to the development of
harmonic analysis on the $p$-adic special orthogonal groups, with potential
applications in $p$-adic quantum mechanics and in the recently proposed
$p$-adic quantum information theory.
