SQL Server 2012 Express is now available to download from this link.
After downloading it and installing it, I tried to integrate it into an existing project in Visual Studio through the Server Explorer, and got the following issue:
This is apparently a known issue, and happens when you right click on "Data Connections" in Server Explorer, and chose "Create New SQL Server Database".
The work around is to create your database using Management Studio (Just the DB, not the actual schema).
Once it has been created, you can then "connect" to the database in Visual Studio's Server Explorer:
This will allow you to connect to the database created in Management Studio without any issues.
If you didn't change it during installation, your instance name will be ".\SQLEXPRESS".