Not necessarily. A test strategy is not a document but it can be documented. I would suggest not to write a test plan if you do not have to. In many context you do not need a test plan. A test plan is often full of fluff that nobody reads anyway.
You always will have a test strategy, even if you are not aware of it. It is in your head when every you test. Your mental model of what to test and how is a test strategy. As soon as you need to communicate it, it might be handy to document parts of it. I like mind maps and a risk-task list amongst other things to help me talk about my test strategy. These visualisations of the product and my test coverage create insight and overview, to help me determine my next steps.
So I would recommend to talk to your team and other stakeholders about how to communicate your test strategy. You’ll find out what helps you and what not.