  In this paper we study the second cohomology space for the $n$-th
Schr\"{o}dinger algebra $\mathfrak{sch}_n$. We prove that the second cohomology
space is vanishing for $n \geq 3$ and show that $\dim(H^2(\mathfrak{sch}_2,
\mathfrak{sch}_2)) =2.$ Moreover, we investigate the factor algebra of the
Schr\"{o}dinger algebra by its center and determine the second cohomology group
of this factor algebra.
