We prove a bounded decomposition for higher order Hankel forms and characterize the first order Hochschild cohomology groups of the disk algebra with coefficients in the space of bounded Hankel forms of some fixed order. Although these groups are non-trivial, we prove that every bounded derivation is inner and necessarily implemented by a Hankel form of order one higher. In terms of operators, this result extends the similarity result of Aleksandrov and Peller. Both of the main structural theorems here rely on estimates involving multilinear maps on the $n$-fold product of the disk algebra and we obtain several higher order analogues of the factorization results due to Aleksandrov and Peller. 2000 Mathematics Subject Classification: 47B35, 46E15, 46E25.