```sh
openssl ecparam -genkey -name prime256v1 | openssl ec -out private.key
```

A response similar to this one should be displayed:

```
read EC key
writing EC key
```

Alternatively, use the following command to generate a private ECDSA key protected by a password:

```sh
openssl ecparam -genkey -name prime256v1 | openssl ec -aes256 -out private.key -passout pass:PASSWORD
```