How to avoid being asked passphrase each time I push to Bitbucket

You need to use an ssh agent. Short answer: try

$ ssh-add

before pushing. Supply your passphrase when asked.

If you aren't already running an ssh agent you will get the following message:

Could not open a connection to your authentication agent.

In that situation, you can start one and set your environment up thusly

eval $(ssh-agent)

Then repeat the ssh-add command.

It's worth taking a look at the ssh agent manpage.

A way to solve this is with ssh-agent and ssh-add:

$ exec ssh-agent bash
$ ssh-add
Enter passphrase for ~/.ssh/id_rsa: 

After this the passphrase is saved for the current session. and won't be asked again.

I use Keychain for managing ssh keys. It is also available in Debian and so presumably Ubuntu with

apt-get install keychain

Here is the Debian keychain package page. As you can see, the project is not very active, but works for me. I also commented a bit about this in another answer here