Hello once a user is given full permission to a mailbox he can choose it and send it from it without any other configuration right ? I am having a problem as when I am choosing this e-mail in the From field and I send it, it comes back as undeliverable due to missing permissions.

Full Mailbox permission is not the same as SEND AS or SEND on BEHALF permissions, so you need to give SEND AS permissions to the user. What version of Exchange is this? For Exchange 2007/2010: http://www.shudnow.net/2007/08/12/send-on-behalf-and-send-as/

it is exchange 2010, so apart from full permission I also need to add send as permission ? I thought full permission would cover all

From Exchange 2003 onwards even if you are recipeint administrator or higher you need to assign send as permission seperately The link may be helpful http://technet.microsoft.com/en-us/library/bb676368.aspx