We consider a unit speed curve $\alpha$ in the Euclidean $n$-dimensional space $\mathbb{E}^n$ and denote the Frenet frame of $\alpha$ by $\{V_1,..., V_n\}$. We say that $\alpha$ is a cylindrical helix if its tangent vector $V_1$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms of their curvatures.