Yes, just stay logged in with the same account you signed up for.
Before reinstalling it, try Run the Troubleshooter: Select the Start button and select Settings > Update and security > Troubleshoot and from the list select Windows Store Apps > Run Troubleshooter. Restart your computer and see if the problem persists.
When purchasing an App through the Microsoft Store, it is linked to your Microsoft account and whenever necessary, just look at your order history or, on the page designed to purchase the product, being logged in with your account, it will not show the Buy option.
For more information, check out the article for information on Viewing your Microsoft Store order history.
If you have any questions on the way, I will be available.
Thanks.
Douglas R