If I understand your question correctly, I wouldn't try to store the connection for later reuse. Instead I would let connection pooling to take care of caching the connection objects. Have a look at
Features of Oracle Data Provider for .NET[
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What comes to the connection string itself, you can store it for quick access for example in a configuration file, in a single, static string variable etc.
For example, have a look at
Four Ways to Read Configuration Setting in C#[
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