Send mail from an additional Exchange address or alias

I’ve got a single Exchange mailbox which holds multiple email addresses. I can receive mail sent to any of these addresses just fine, but I cannot find a way to choose which address to send out with.

I’ve tried setting the From field with another address but that got automatically replaced with my default address or a bounce message saying that I don’t have permission to do so.

How can I send out a message with one of my mailbox aliases?