It sounds like you want a provisioning tool such as ansible, saltstack, chef or puppet.
Ansible works in a very similar way to what you want, you run it locally and it sshs into remote servers to run commands (there is no need to install an agent on the remote servers like the other products listed above). For example given a list of servers in
/etc/ansible/hosts you can run
ansible all -a "/bin/echo hello" to execute the given command on all servers. Full guide here.
But in addition to this you can start to craft playbooks of useful commands that let you rerun them on hosts in the future and work towards automating management of your infrastructure.
Alternatively you can use commands like pssh for running batch commands in parallel or using the synchronize-panes feature of tmux to run commands interactively across multiple hosts.