  The $SO(N)$ Yang-Mills gauge theory is concerned since it can be used to
explore the new theory beyond the standard model of particle physics and the
higher dimensional loop quantum gravity. The canonical formulation and loop
quantization of $SO(N)$ Yang-Mills theory suggest a discrete $SO(N)$
holonomy-flux phase space, and the properties of the critical quantum algebras
in the loop quantized $SO(N)$ Yang-Mills theory are encoded in the symplectic
structure of this $SO(N)$ holonomy-flux phase space. With the $SO(D+1)$ loop
quantum gravity as an exemplification of loop quantized $SO(N)$ Yang-Mills
gauge theory, we introduce a new parametrization of the $SO(D+1)$ holonomy-flux
phase space in this paper. Moreover, the symplectic structure of the $SO(D+1)$
holonomy-flux phase space are analyzed in terms of the parametrization
variables. Comparing to the Poisson algebras among the $SO(D+1)$ holonomy-flux
variables, it is shown that the Poisson algebras among the parametrization
variables take a clearer formulation, i.e., the Lie algebras of $so(D+1)$ and
the Poisson algebras between angle-length pairs.
