  We construct infinite-dimensional analogues of finite-dimensional simple
modules of the nonstandard $q$-deformed enveloping algebra
$U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for
the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper
we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the
classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide
rational matrix coefficients for these infinite-dimensional modules of both
$U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with
rationalized formulas to embed the respective algebras into skew group algebras
of shift operators. Casimir elements of $U_q'(\mathfrak{so}_n)$ were given by
Gavrilik and Iorgov, and we consider the commutative subalgebra $\Gamma\subset
U_q'(\mathfrak{so}_n)$ generated by these elements and the corresponding
subalgebra $\Gamma_1\subset U(\mathfrak{so}_n)$. The images of $\Gamma$ and
$\Gamma_1$ under their respective embeddings into skew group algebras are equal
to invariant algebras under certain group actions. We use these facts to show
$\Gamma$ is a Harish-Chandra subalgebra of $U_q'(\mathfrak{so}_n)$ and
$\Gamma_1$ is a Harish-Chandra subalgebra of $U(\mathfrak{so}_n)$.
