This paper presents the so-called symmetric high-order microplane (SYHOM) model for the simulation of damage localization and size effect in quasi-brittle materials. Contrarily to its predecessor, the high order microplane (HOM) model, SYHOM is formulated without rotational degrees of freedom, does not require couple stresses, and solves stability issues created by the antisymmetric components of stress and high order stress tensors. Furthermore, the formulation features variable, damage-dependent localization limiter and nonlocal characteristic length that allows reproducing correctly size and shape of the fracture process zone during the entire softening process: from crack initiation to the formation of a stress-free fracture. The paper highlights the implementation of SYHOM via isogeometric analysis with quadratic shape functions and presents several numerical simulations to demonstrate SYHOM's ability to simulate damage evolution during softening. Finally, SYHOM is validated by simulating experimental data relevant to the size effect on structural strength of notched concrete samples.