  We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb
R^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb {R}^n)$, the general
Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling
type, in terms of decay estimates for the Fourier coefficients of their
elements with respect to eigenfunction expansions associated to normal globally
elliptic differential operators of Shubin type. Moreover, we show that the
eigenfunctions of such operators are absolute Schauder bases for these spaces
of ultradifferentiable functions. Our characterization extends earlier results
by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361-4368) for Gevrey
weight sequences. It also generalizes to $\mathbb{R}^{n}$ recent results by
Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds.
