  In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ in
the quantum groups $SU_{q}(N)$ to transform the conjugate representation into
the standard form so that we are able to compute the explicit forms of the
important quantities in the bicovariant differential calculus on $SU_{q}(N)$,
such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ and the
$q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariant
condition we define the generalized $q$-deformed Killing form and the $m$-th
$q$-deformed Chern class $P_{m}$ for the quantum groups $SU_{q}(N)$. Some
useful relations of the generalized $q$-deformed Killing form are presented. In
terms of the $q$-deformed homotopy operator we are able to compute the
$q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $dQ_{2m-1}=P_{m}$,
Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge
covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived.
