Creating a generic class pool where you give the generic parameter and get a generic object that used that parameter

Question

Goal

I have a generic class GenericClass<T> and I want to pool instances.
I'm interested in seeing if I can get the syntax:

MyGenericPool = new GenericPool<GenericClass>();
// Or maybe it's MyGenericPool = new GenericPool<GenericClass<>>();

GenericClass<TGenericParam> GenericClassInstance =
    MyGenericPool.Get<TGenericParam>();

^{(My understanding of generics says, no I can't, don't be silly the syntax doesn't exist / wouldn't work, but I'm intested in what others think).}

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Showing my workings

I'm a bit doubtful as from my understanding the types GenericClass<string> and GenericClass<int> aren't really related from the type system's point of view.

Now, I realise that I can get close, i.e.:

GenericClass<TGenericParam> GenericClassInstance =
    GenericPool.Get<GenericClass<TGenericParam>>();

and then have the GenericPool just store a Dictionary<Type, ObjectPool<object>> somewhere.
I'm interested in seeing if I can avoid having to do that. I don't want to have to specify the generic type every time when, as the caller, i'm only changing the generic type parameter. I'd also like to be able to enforce (compile time) that all objects going into my GenericObjectPool<T> are of a set generic type (T<>).

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I think the problem stems from not being able to treat a generic type parameter as being generic its self. If I could do that (can I already??) then maybe something like the below might work:

public class GenericClassPool<TGeneric> where TGeneric : class<>
{
    private readonly Dictionary<Type, object> objectPools = new Dictionary<Type, object>();


    private void EnsureObjectPoolExists<TGenericParam>()
    {
        if (!objectPools.ContainsKey(typeof(TGenericParam)))
        {
            objectPools.Add(typeof(TGenericParam), new ObjectPool<TGeneric<TGenericParam>>(() => Activator.CreateInstance(typeof(TGeneric<TGenericParam>)) as TGeneric<TGenericParam>));
        }
    }

    private ObjectPool<TGeneric<TGenericParam>> GetPool<TGenericParam>()
    {
        EnsureObjectPoolExists<TGenericParam>();
        return (objectPools[typeof(TGenericParam)] as ObjectPool<TGeneric<TGenericParam>>);
    }

    public void Add<TTypeParam>(TGeneric<TGenericParam> obj)
    {
        EnsureObjectPoolExists<TTypeParam>();

        GetPool<TGenericParam>().Add(obj);
    }

    public TGeneric<TGenericParam> Get<TGenericParam>()
    {
        return GetPool<TGenericParam>().Get() as TGeneric<TGenericParam>;
    }
}

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Question

Can I get the syntax I want (at the top)? If not, how close can I get?