We define a unified trace form Ï„ on the cyclotomic Hecke algebras
Hn,K​ of type A, which generalize both Malle-Mathas' trace form
on the non-degenerate version (with Hecke parameter Î¾î€ =1) and
Brundan-Kleshchev's trace form on the degenerate version. We use seminormal
basis theory to construct a pair of dual bases for Hn,K​ with
respect to the form. We also construct an explicit basis for the cocenter
(i.e., the 0th Hochschild homology) of the corresponding cyclotomic Schur
algebra, which shows that the cocenter has dimension independent of the ground
field K, the Hecke parameter ξ and the cyclotomic parameters
Q1​,⋯,Qℓ​