variant of Heisenberg group

Question

If $A$ is skew-symmetric matrix of $M_n$, it seems to me that $G=\mathbb{R}^n \times \mathbb{R}$ equipped with the product $$ (s,t) \cdot (s', t') = \bigg(s + s', t + t' + \frac{1}{2} \langle s, A s'\rangle\bigg) $$ is a group. Here $\langle \cdot, \cdot \rangle$ is the canonical scalar product on $\mathbb{R}^n$. Out by curiosity, does it have a name? Is it studied somewhere ?