Yes, you can!
We have tested several services regarding this use case and found an optimal solution for you: the Matterbridge service enables you to bridge between various protocols. Following this link, you can get the guideline for installation.
Please notice that data cannot be bridged on some occasions. For instance, we bridged between Matrix and Mattermost for testing, and transferring private messages is impossible. Besides, the message type unique to the platform cannot be bridged either.
The usage of Matterbridge is based on your individual use scenario, so we will not set it up as a central service. However, you are very welcome to contact us if you need any technical support in configuration.