Calculating transposes for elements of $\operatorname{End}(\mathbb{R}^{\infty})$.

Question

Here, $\mathbb{R}^{\infty}$ is the vector space of infinite sequences where only finitely many elements are nonzero. I could easily calculate this for a finite-dimensional vector space, but now when working with infinite dimensional vector spaces it has gotten really complex. I haven't formally learnt this and the summaries on other stack exchange posts and the wikipedia summary are quite confusing and seem to be very general. I was wondering if there is a nice way of calculating transposes for elements of $\operatorname{End}(\mathbb{R}^{\infty})$, which is hopefully simpler than the general case.