Given a Hilbert space \({\mathcal H}\) with inner product \(\left< \cdot, \cdot \right>\) and its induced norm \(\|\cdot\|\), the Cauchy-Schwarz inequality asserts \[ | \langle a, b\rangle | \le \| a\| \|b\|, \quad a, b\in {\mathcal H}. \] The equality holds if and only if \(a\) and \(b\) are linearly dependent. The Ostrowski inequality is a refinement of the Cauchy-Schwarz inequality in the finite-dimensional case. M. Fujii et al. [Sci. Math. 2, No. 2, 215–221 (1999; Zbl 0961.47006)] extended the Ostrowski inequality to a vector version for Hilbert space, which is not symmetric with respect to the vectors \(a\) and \(b\). Thus, the authors give a symmetric generalization of the Ostrowski inequality in this paper. They also provide a two variable extension of the Ostrowski inequality. Furthermore, based on the matrix version of this symmetric generalization, the authors provide matrix inequalities involving the matrix geometric mean of two positive semidefinite matrices.