Large Vector Notation

If without writing any components perhaps for $\mathbf v \in \mathbb R^n$ you could write $\mathbf u = \mathbf v/e_i \in \mathbb R^{n-1}$, as a ''qoutient'', where $e_i$ is the $i$-th standard basis vector in $\mathbb R^n$. I hope you know quotient spaces.

In some textbooks hats indicate the omission of elements $$ \mathbb R^{n}/\text{span}\{e_i\} \cong \mathbb R^{n-1} \ni \mathbf v/e_i = (v_1,\dots,\widehat{v_i},\dots,v_n)^T = (v_1,\dots,v_{i-1},v_{i+1},\dots,v_n)^T $$

If you need it a lot in one text, I think defining an ad hoc notation could be useful. For example, I would consider using something simple like $\mathbf{v}^{(i)}$, $ \mathbf v^i$, or something similar. Otherwise, the comments and the other answer already give the standard notations I can think of.