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In this paper, we investigate rotational hypersurfaces family in $n$-dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying theGauss map $\mathcal{G}$ of this family with respect to the operator$\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. Theoperator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known byvarious names, including the Cheng--Yau operator in certain cases.Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applyingthe operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish aclassification theorem. This theorem establishes a connection between the$n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through theequation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$.

Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$