It is known that the Stokes operator is not well-defined in $L^q$-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see \cite{FKS1}, to the $n$-dimensional one, $n\geq 2$, replacing the space $L^q, 1\ltq\lt\infty$, by $\s{L}^q$ where $\s{L}^q = L^q\cap L^2$ for $q\geq 2$ and $\s{L}^q = L^q+L^2$ for $1\ltq\lt2$. In particular, we show that the Stokes operator is well-defined in $\s{L}^q$ for every unbounded domain of uniform $C^{1,1}$-type in $\R^n$, $n\geq 2$, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.