We can send Bitcoin without using an exchange by downloading desktop wallet or mobile wallet, keeping the fund there and moving it if you wanted to send a transaction. They are free and available online. Among these wallets, I am fond of using electrum lightweight wallet.
electrum is very simple to use bitcoin, I also use it, I do not recognize wallet desktop or wallet that need to be downloaded because it is very susceptible to hacking activity. There are many ways to conduct transactions other than by using a exchange wallet. and one of the best is to use the web-wallet.