Let $V$ be a finite dimensional vector space over $\mathbb{R}$ of dimension $n$.
The vector space $\underbrace{V \oplus V \oplus \cdots \oplus V}_k$ has dimension $kn$, and the vector space $\underbrace{V \otimes V \otimes \cdots \otimes V}_k$ has dimension $n^k$.
Is there a similar construction (that does not depend on a basis of $V$), that gives a vector space of dimension $2^n$?