I don't think you can use symmetric key because it contradicts the purpose of signing. The main purpose of signing is to ensure authenticity and non-repudiation. Using private key ensures that it is from the intended sender and not any one else, given that private key is not compromised or stolen. Check here for the full process.
By using symmetric key, there is chance that the same key is with another person also(shared for some other transaction by sender). So, you cannot be sure that it came from the original sender or the someone with the same symmetric key, which contradicts the purpose of signing.
