This is not possible in game. However, if you use an item manager you can create loadouts. For example, check out Destiny Item Manager here:
https://app.destinyitemmanager.com
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is that basically like the Destiny companion app?
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It's similar, but has existed for longer than Bungie's gear manager in the companion app. It uses Bungie's OAUTH system so you are not entering your username and password into a foreign website.