In McDuff and Salamon's Introduction to symplectic topology. They mention that there is a contact structure on $\mathbb{R}^n \times S^{n-1}$ which extends the standard one on $\mathbb{R}^{2n-1}$.
To be more precise, the standard contact structure on $J^1\mathbb{R}^{n-1} = \mathbb{R}^{2n-1} = \mathbb{R}^{n-1}_x \times \mathbb{R}^{n-1}_y \times \mathbb{R}_z$ is given by the contact form $\alpha_0 = dz - \sum_i y_i dx_i$, and the contact form $\alpha_1$ on $S^\ast \mathbb{R}^n = \mathbb{R}^n \times S^{n-1}$ is given by restricting $ \sum_i a_i du_i$ where $(u,a)$ is the standard coordinate for the cotangent bundle $T^\ast \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$ to $S^\ast \mathbb{R}^n$.
What's an explicit formula of the map?