Avoid angle brackets in default template

You can use typedef...

typedef tt<> tt_;

And then simply use tt_.

... if I want to use the class ...

This is a common source of confusion. A class template is not a class, but a template from which classes are generated. The angle brackets is what tells the compiler that you want to generate a class out of the class template with the given template arguments, without the angle brackets what you have is a template.

template <typename T = int>
struct TemplateClass { /*...*/ };

template <template <typename>  class T>
void f() {
   T<int> t; // ...
}
template <typename T>
void g() {
   T t; // ...
}

f<TemplateClass>();     // Accepts a template with a single type argument
g<TemplateClass<> >();  // Accepts a type, that can be generated out of the template

The language does not allow the coexistence of a template and a type with the same name in the same namespace, so the answer is that it cannot be done. You can create a type alias but you will have to give it a different name.