  We define a unified trace form $\tau$ on the cyclotomic Hecke algebras
$\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form
on the non-degenerate version (with Hecke parameter $\xi\neq 1$) and
Brundan-Kleshchev's trace form on the degenerate version. We use seminormal
basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ with
respect to the form. We also construct an explicit basis for the cocenter
(i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur
algebra, which shows that the cocenter has dimension independent of the ground
field $K$, the Hecke parameter $\xi$ and the cyclotomic parameters
$Q_1,\cdots,Q_\ell$.
