Cloud storage has benefited millions of users with its remarkable advantages of economy and flexibility. Since a single cloud service provider is not always reliable and prone to a single point of failure, multi-cloud storage is proposed to enhance data availability. However, cross multiple clouds (cross-cloud) auditing to provide users with the integrity verification of outsourced data also faces many challenges, such as the fact that a large quantity of existing schemes rely heavily on the trusted third-party. Therefore, we propose a decentralized data storage and integrity auditing scheme for multi-cloud scenarios to eliminate the dependence on the third-party, namely BDDR, and an improved version iBDDR with stronger security. First, both BDDR and iBDDR adopt multi-cloud data storage and support batch auditing to greatly save computation costs. Second, our schemes not only get rid of a third-party, but also do not require a dispute arbitration since the outsourced data can be verified by cloud server providers via the homomorphic verifiable tags published on the blockchain. Third, locating the corrupted cloud server providers and accurately recovering the corrupted data at a lower communication and computation costs are supported. Finally, security and performance analyses demonstrate the security and effectiveness of our schemes.