A decomposition theorem for maximum weight matchings
Are you actually looking for a polynomial time algorithm solving maximum weight matching for general graphs, or a specific way to solve it using a kind of decomposition?

In the former case, the following paper gives an $\mathcal{O}\big(n(m + \log n)\big)$ exact algorithm:
H.N.Gabow. Data Structures for Weighted Matching and Nearest Common Ancestors witk Linking. SODA'90 (1990)
And for a $(2/3 - \epsilon)$-approximation algorithm running in $\mathcal{O}(m \log 1/\epsilon)$:
S.Pettie, P.Sanders. A Simpler Linear $2/3 - \epsilon$ Approximation Algorithm for Maximum Weight Matching. Information Processing Letters, vol.91 (2004)