Good day,
Not sure what you mean by connect it with SQL express. My feeling is that you confuse two unrelated application: (1) SQL Server Express and (2) SQL Server Management Studio (SSMS). With that being said, it is possible that you meant to connect from one server to the other using linked server for example. I will answer both options...
Connecting from SSMS to AWS SQL Server
This is well documented in the official AWS system as you can see here. Basically you need to copy your server endpoint from your account.
If this is what you meant so you must understand that SSMS is a client application and had nothing to do with SQL Server (meaning it is not part of SQL Server but totally separate application, which can be install with or without SQL Server like any unrelated app)
Create linked server from local server to AWS SQL Server
If you do mean to connect using linked server from one server to the other, then:
Step 1: Do the same as above option and get the information to connect from SSMS. We need the Server name, user name, and password.
Step 2: Confirm that you can connect from SSMS first.
Step 3: continue to create linked server, and map your remote username to the local server. For this you can use the bellow queries.
USE [master]
GO
-- create linked server to the remote server
EXEC master.dbo.sp_addlinkedserver
@server = N'<Your Server name come here>', -- this is the same information as you use from SSMS
@srvproduct=N'SQL Server';
GO
-- check your list of servers to confirm that you see the new one
select * from sys.servers
GO
-- Now we need to map your remote login to your new linked server
EXEC master.dbo.sp_addlinkedsrvlogin
@rmtsrvname = N'<Your Server name come here>',
@useself = 'FALSE',
@locallogin = NULL ,
@rmtuser = N'RonenAriely', -- enter your remote user name
@rmtpassword = '<Enter Your remote Password here>'
GO
-- that's it, wee can query the remote server from the local server now
--check that your can execute remote query
SELECT name FROM [<Your Server name come here>].master.sys.databases
GO
I hope this solve your issue :-)