  Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$
from a circular orthogonal ensemble. In this paper, we study moments of a
single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit
values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading
and subleading terms in the asymptotic expansion with respect to a large matrix
size $N$. Our technique is to apply the Weingarten calculus for a
Haar-distributed unitary matrix.
