If your company has enabled the RPC over HTTP Option for the Exchange server then you can use Outlook remotely. If they have not (it is not enabled by default) then you can not use Outlook and you'll need to continue to use Outlook Web Access (that's what the website you are using is called).
In Outlook click on Tools then Accounts. Add a new account. Select Microsoft Exchange and click Next. Select Microsoft Exchange again and select Next. Click on More Settings. On the general tab type in a name for the Account (Exchange, Work, etc).
On the Connection tab check the Connect to Microsoft Exchange using HTTP and click the Exchange Proxy Settings button.
On the new window type in the URL for the Website that you currently use for accessing your mailbox without any folder names. It'll be something like webmail.yourcompany.com or mail.yourcompany.com, etc. Check the connect using SSL only. In the proxy authentication drop down select the NTLM authentication option. Click ok, and the ok again.
On the new account screen you'll need to enter the name of the Exchange server. You can try entering the name of the website site, and in the User Name field enter your username to log onto the web mail. Click the Check Name box and see if it finds your mailbox.
You may need to play with the authentication settings, and you may need to know the actual name of the Exchange server to make this work.
Keep in mind this might not work at all because RCP over HTTP is disabled. Your best bet would be to ask your company mail admin or help desk if using Outlook to access Exchange from home is supported or not. They would be able to provide instructions specific to your environment.