Text-based conversations between users with searchable history in Microsoft Teams Free
Teams Free does not require any Microsoft 365 subscription. You should be able to sign in to Teams with a personal Microsoft account or Skype Name. However, if your Skype Name starts with the "live:" prefix, then this ID cannot be used as Login ID for Teams.
Uninstall any currently installed Teams application.
https://support.microsoft.com/en-us/teams/troubleshooting/uninstall-microsoft-teams
Install the latest Teams app using this installation site.
https://www.microsoft.com/en-us/microsoft-teams/download-app
Sign in to Teams with exactly the same account as you have used with Skype.