raster - How many polygons does a single polygon intersect with?

I currently possess a shapefile that consists of >5000 polygons. I would like to know, for each polygon, how many other polygons it intersects with and the names of each polygon it intersects with?

My output should be something like this:

Based on this previous question and @JeffreyEvans suggestion on using gIntersects, I have tried the following below. But, the process has been running continuously without stopping for the last 5 hours. Any suggestions?

Here's the code I ran in R:

require(rgdal)
library(maptools)

library(rgeos)
library(GISTools)
Poly <- readOGR("C:\\Users\\rameshv\\Terrestrial_dissolved_multipart.shp")
Poly_join <- gIntersects(Poly)

EDIT

I was able to solve part of my own question. However, I am still unclear as to write a list of comma separated names on each row. As of now, I have a loop that read in every shapefile, checks for an Intersection with a large Spatial Polygons Data Frame and prints out a dataframe of and number of polygons it intersects with.

library(rgdal)
library(GISTools)
library(sp)

library(raster)

##List all the shapefiles
shapes <- list.files(path =    "C:\\Users\\rameshv\\Desktop\\Output_All\\", pattern=".shp$", full.names=TRUE)
head(shapes)

##Read in the big spatial polygons dataframe - this contains ALL MAMMALS
Big <-readOGR("C:\\Users\\rameshv\\Desktop\\Ranges_Overlap\\multipart.shp")

for (i in 1:length(shapes)){

    a <- readOGR(shapes[i])
    o <- over(a, Big,returnList = T)
    s <- o[[1]]
    count <- length(s$binomial)-1
    a$binomial <- as.character(a$binomial)
    results <- data.frame(a$binomial,count)
 write.table(results,file="C:\\Users\\rameshv\\Desktop\\overlapfile_Final.csv", row.names=F,append = TRUE, col.names = F, sep = ",")
}

Any thoughts?

Answer

Now with sf you could do

st_intersection(shapes, shapes)

For example

library(sf)
example(st_read)
x <- st_buffer(nc[1:10, ], dist = 0.1) ## make sure some overlap
st_intersects(x, x)
 Sparse geometry binary predicate list of length 10, where the predicate was `intersects'

 1: 1, 2
 2: 1, 2, 3
 3: 2, 3, 10
 4: 4, 7, 8
 5: 5, 6, 8, 9
 6: 5, 6, 8
 7: 4, 7, 8
 8: 4, 5, 6, 7, 8
 9: 5, 9
 10: 3, 10