Victor Ivanidze
My understanding is that the only way to send from multiple domains is to have a shared mailbox for each domain and then send from the appropriate shared mail box.
Although it is on Microsoft's roadmap, you cannot send from an email alias. You can define email aliases for multiple domains on your user account and receive emails sent to those aliases, but you currently cannot send from those aliases.
To get around this, I have created a shared mailbox for my personal email domain. I can then send from that shared mailbox and the message goes from <me>@<personaldomain>.
I am limited to sending from only one address at that domain, but its an improvement on the standard limited behaviour.