@Abby,
No, you can't grant "Send As" on Exchange servers or the organization object to get the same effect as granting "Send As" on user accounts.
For the Exchange organization and server objects, "Receive As" is the functional equivalent of "Full Mailbox Access." Getting this permission on a database gives you access to all the contents of the database--including mailboxes and folders.
So you'd think "Send As" on a database or on the whole organization would give you rights to send as any mailbox in a database, but it doesn't. It actually gives you the right to send as *the database object* (lot of good that does you).
Send As must be granted on the user account or on a container that holds user accounts, not on a database or server.
To grant Send As for all users inside a container, you must use the Advanced button on the Security properties for the container. From there, you can grant an account Send As for all User Objects in the container.